Methods and Apparatuses for Advanced Multiple Variable Control with High Dimension Multiple Constraints

ABSTRACT

A method and apparatus for multiple variable control of a physical plant with high dimension multiple constraints, includes: mathematically decoupling primary controlled outputs of a controlled physical plant from one another and shaping the pseudo inputs/controlled outputs desired plant dynamics; tracking primary control references and providing pseudo inputs generated by desired primary output tracking for selection; mathematically decoupling constraints from one another; mathematically decoupling constraints from non-traded off primary controlled outputs of the controlled physical plant; shaping the pseudo inputs/constraint outputs desired plant dynamics; tracking constraint control limits; providing pseudo inputs generated by desired constraint output tracking for selection; selecting the most limiting constraints and providing the smooth pseudo inputs for the decoupled primary control; and controlling the physical plant using the decoupled non-traded off primary controlled outputs and the decoupled selected most limiting constraints.

CROSS REFERENCE TO RELATED APPLICATIONS

The current application claims priority to U.S. Provisional ApplicationSer. No. 61/597,316, filed Feb. 10, 2012, the entire disclosure of whichis incorporated herein by reference.

BACKGROUND OF THE INVENTION

The present disclosure pertains to control system design and operationwhen the controlled system includes multiple control targets withmultiple constraints.

In a control system having more than one primary target to be controlledfor multiple primary control objectives, such as thrust, fanoperability, core operability, etc. (for a jet engine, for example), thecontrol system will have multiple inputs and multiple outputs tocontrol. Such a control system should address the challenge ofmulti-variable control with multiple constraints, particularly when theprimary control objectives have high transient and dynamic requirements.The challenge fundamentally is a coordinated control to maintain primarycontrol objectives as much as possible while enforcing a selected set ofactive constraints that can satisfy all potentially active constraints.

Traditionally, single-input-single-output (SISO) control is used for oneprimary control objective—for example in a gas turbine engine, fan speedonly. The concerned constraints are converted to the control actuatorrate—fuel rate, respectively, the constraint demanding most fuel rate isselected as most limiting constraint and enforced. Here, there is anassumption that fuel rate is always proportional to fan speed change,and fan speed changes always align up and dominate the thrust responseand operability. This may be true in many operating conditions, but itis not true for certain operating conditions, such as supersonicoperating area for conventional engine applications, not to mentionnon-conventional engine applications, such as powered lift operation.

Multiple constraints may be in one subset only, that is, at same time,only one primary controlled output needs to be traded off. There arecases, however, in which multiple constraints are in two or more subsetsthat require two or more primary controlled outputs to be traded off.Certainly, at most, the number of the subsets should be equal to thenumber of the primary control handles. For example, in the gas turbineengine example, if both “maximum core speed” and “maximum exhausttemperature” constraints are active, it may be necessary to trade offboth primary controlled outputs, “fan speed” and “pressure ratio,” forbetter thrust and operability performance, while enforcing both the“maximum core speed” and “maximum exhaust temperature” constraints. Itis a challenge to control multiple variables with higher dimensionmultiple constraints.

Previous approaches to solve this problem have either greatlyoversimplified the problem or added substantial complexity. Theoversimplified approach ignored fundamental confounding in therelationships between the controlled plant inputs and the performancetrade-off and control mode selection decisions that must be made. Thislimited its applicability to certain 2×2 multi-input-multi-output (MIMO)systems, and does not represent a robust solution for higher dimensionMIMO systems. The overly complicated approaches coupled the constraintcontrol with the primary control, usually lost expected controlobjectives priority, and sacrificed the physical meaning, robustness,deterministicness, and maintainability of the control solution.

BRIEF DESCRIPTION OF THE INVENTION

The present disclosure provides a control system design methodology thatincorporates a simple, deterministic, robust and systematic solutionwith explicit physical meaning for the advanced multi-variable control,with high dimension multiple constraint problems where the trade-offprimary control outputs are pre-determined based on plant physics (forexample, engine, or other plant characteristics depending upon theapplication). The disclosed methodology provides a fundamental solutionfor the problem of MIMO control with multiple constraints and/ormultiple high dimension constraints, it follows that the resultedsolution well coordinates the multi-variable control with the selectedmultiple active constraints enforcement such that, when not constrained,the primary multi-variable control has its optimized performance asdesigned; when constrained, the proper most limiting active constraintsare correctly selected and naturally enforced by replacing thepre-determined trade-off primary control outputs, respectively. If themost limiting constraints are enforced, then the rest of the constraintscan be automatically satisfied. Together they make the overall systemstill have desired primary control performance while running under theenforced constraints, and the traded-off primary control outputs havenatural fall-out.

Consider the Multiple Variable Control with Multiple Constraints as awhole space. One subspace is a class with only one subset of constraintsto be active and only one primary control output to be traded off—thisis single-dimension multiple constraints case. The rest of the space iswith two or more subsets of constraints to be active and two or moreprimary controls to be traded off—this is high dimension multipleconstraints case. The control system design methodology provided by thisdisclosure is not only for single dimension multiple constraints casebut also for high dimension multiple constraints case.

The control system design methodology provided by this disclosureresults in a simple physics based selection logic, and a mathematicallydecoupled primary control with decoupled constraint control. That is,the primary controlled outputs are mathematically decoupled from oneanother, the selected constraints in control are mathematicallydecoupled from one another, and the selected constraints aremathematically decoupled from primary controlled outputs. It followsthat each decoupled control target can be designed viasingle-input-single-output (SISO) control approaches for its specificperformance requirements.

According to the current disclosure, an embodiment of a control systemfor a physical plant (such as, for example and without limitation, a gasturbine engine control, flight control, satellite control, rocketcontrol, automotive control, industrial process control) may include: aset of control reference signals; a set of controlled output feedbacksignals from the physical plant; a multiple input multiple output (MIMO)primary decoupling controller providing control command derivatives tothe integral action (and enabling the shaping of desired robust controlof the primary control outputs); a set of SISO lead/lag controllers thatcan extend the bandwidths of decoupled primary SISO loops, respectively;a set of decoupled SISO controllers for controlled outputs tracking thatreceive primary controlled output tracking errors and provide desiredpseudo inputs, respectively; a multiple input multiple output (MIMO)constraint decoupling controller decoupling the constraints from oneanother, decoupling the constraints from the non-traded off primarycontrolled outputs, and providing pseudo inputs based on the desiredconstraint responses for the Selection Logic (introduced below); a setof decoupled SISO controllers for constraint outputs tracking thatreceive constraint output tracking errors and shape desired constraintresponses, respectively; a Selection Logic which compares the pseudoinputs generated by each subset of constraints and the pseudo inputgenerated by the primary controlled output associated with that subset,selects the most limiting constraint for that subset, and determines thefinal pseudo inputs to go into the SISO Lead/Lag and MIMO PrimaryDecoupling Controller. With such an architecture, the integral actionbecomes a set of common SISO integrators shared by primary control andconstraint control.

According to the current disclosure, a control system for a physicalplant, includes: an integral action control unit providing controlsignals for a physical plant; a multiple-input-multiple-output (MIMO)primary decoupling controller providing control command derivatives tothe integral action control unit and thereby forming at least adecoupled controlled plant; and a multiple-input-multiple-output (MIMO)constraint decoupling controller decoupling constraint outputs from thephysical plant and providing pseudo inputs to the above decoupledcontrolled plant. In a more detailed embodiment, a selection logicsection for selecting pseudo inputs for the primary decouplingcontroller from those pseudo inputs calculated by: 1) the MIMOconstraint decoupling controller and the constraint tracking controller;2) the primary MIMO decoupling controller and the output trackingcontroller. In a further detailed embodiment, the control system furtherincludes a set of decoupled single-input-single-output (SISO) controlledoutput tracking controllers receiving controlled output tracking errorsignals and providing pseudo input signals to the decoupled controlledplant. In a further detailed embodiment, the selection logic comparesthe pseudo inputs from the MIMO constraint decoupling controller and thepseudo input signals from the SISO controlled output trackingcontrollers and selects at least one most limiting constraint todetermine pseudo inputs to the (SISO) lead/lag controllers.Alternatively, or in addition, the selection logic compares the pseudoinputs from the MIMO constraint decoupling controller and the pseudoinput signals from the SISO controlled output tracking controllers andselects at least one most limiting constraint to determine pseudo inputsto (MIMO) primary decoupling controller.

In an embodiment, the control system further includes a set ofsingle-input-single-output (SISO) lead/lag controllers to extend thebandwidths of decoupled primary SISO control loops, providing v-dot-starto the primary decoupling controller. Alternatively, or in addition, theMIMO constraint decoupling controller decouples the constraints from thenon-traded off primary controlled outputs by rejecting non-traded offprimary controlled outputs as known disturbance inputs. Alternatively,or in addition, the MIMO constraint decoupling controller decouplesconstraint outputs from one another, and decouples the constraints fromthe non-traded off primary controlled outputs. Alternatively, or inaddition, the control system further includes a set ofsingle-input-single-output (SISO) constraint output tracking controllersthat receive constraint output tracking errors from the physical plantand shape desired constraint responses based on the MIMO constraintdecoupling controller. Such constraint output tracking errors may bedetermined, at least in part, based upon the differences betweenpredetermined constraint limits and constraint outputs.

According to the current disclosure a method for multiple variablecontrol of a physical plant with not only multiple inputs and multipleoutputs but also high dimension multiple constraints, includes the stepsof: decoupling the multiple primary controlled outputs from one another,the step of decoupling the multiple primary controlled outputs uses amulti-input-multi-output (MIMO) primary decoupling controller;decoupling the multiple constraints from one another and decoupling themultiple constraints from non-traded off primary controlled outputs, thestep of decoupling the multiple constraints involves amulti-input-multi-output (MIMO) constraint decoupling controller; andproviding pseudo inputs, where the pseudo outputs generated byconstraints are comparable to the pseudo outputs generated by theprimary controlled outputs, to the MIMO primary decoupling controller.The method further includes a step of selecting the most limitingconstraint(s) for the MIMO primary decoupling controller; where the stepof selecting the most limiting constraint includes the step of comparingthe pseudo inputs generated by given subsets of constraints and thepseudo input generated by the primary controlled output associated withthat subset, and selecting the most limiting constraint based, at leastin part, on those comparisons. The MIMO primary decoupling controllermay provide decoupled control using dynamics inversion. The method mayfurther include a step of extending the bandwidths of decoupled primarycontrol loops using a set of single-input-single-output (SISO) lead/lagcontrollers upstream of the MIMO primary decoupling controller. And thestep of decoupling the multiple constraints from non-traded off primarycontrolled outputs includes a step of rejecting the non-traded offprimary controlled outputs as known disturbance inputs.

According to the current disclosure a method for multiple variablecontrol of a physical plant with not only multiple inputs and multipleoutputs but also high dimension multiple constraints, includes the stepsof: mathematically decoupling primary controlled outputs of a controlledphysical plant from one another; mathematically decoupling constraintsfrom one another; mathematically decoupling selected constraints fromnon-traded off primary controlled outputs of the controlled physicalplant; and controlling the physical plant using the decoupled primarycontrolled outputs and/or the decoupled selected constraints which aredecoupled from the non-traded off primary controlled outputs. Such amethod further includes a step of selecting one or more most limitingconstraints.

Development of the advanced multiple variable control with highdimension multiple constraints will now be introduced and discussed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram representation of a control systemarchitecture which can be multiple variable control with high dimensionmultiple constraints or single variable control with single dimensionmultiple constraints embodiment according to the current disclosure;

FIG. 2 is a block diagram representation of an exemplary implementationof the exemplary control system architecture according to the currentdisclosure;

FIG. 3 is a block diagram representation of an exemplary commonsingle-input-single-output (SISO) integrator for use with the currentembodiments;

FIG. 4 is a diagram representation of selection logic for a onedimension constraint set following a min/max selection principle;

FIG. 5 is a diagram representation of selection logic for a onedimension constraint set following a different min/max selectionprinciple; and

FIG. 6 is a flow-chart representation of an exemplary selection logicprocess for high dimension constraint sets according to the currentdisclosure.

DETAILED DESCRIPTION

The present disclosure provides a control system design methodology thatincorporates a simple, deterministic, robust and systematic solutionwith explicit physical meaning for the advanced multi-variable control,with high dimension multiple constraints problems where the trade-offprimary control outputs are pre-determined based on plant physics andperformance requirements (for example, engine, or other plantcharacteristics depending upon the application). The solution wellcoordinates the multi-variable control with the selected multiple activeconstraints enforcement such that, when not constrained, the primarymulti-variable control has its optimized performance as designed; whenconstrained, the proper most limiting active constraints are correctlyselected and naturally enforced by replacing the pre-determinedtrade-off primary control outputs, respectively. If the most limitingconstraints are enforced, then the rest of the constraints can beautomatically satisfied. Together they make the overall system stillhave desired primary control performance while running under theenforced constraints, and the traded-off primary control outputs havenatural fall-out.

When the primary multiple variable control is not constrained, it shouldrun to its desired performance. When the primary control is constrained,the constraint control should keep the most limiting constraints stayingwithin their limits; and at the same time, since constraint control usessome or all primary control handles, the primary control should betraded off in an acceptable way while the intended non-traded-off partof primary control should be not impacted by enforcing the most limitingconstraints.

Based on the above control design goals, first, the primary multiplevariable is designed to have decoupled input/output mapping. Thenmultiple constraint control is designed based on the new controlledplant resulted from the primary control design such that the constraintcontrol not only decouples constraints from one another, decouplesconstraints from the non-traded off primary controlled output(s), alsoprovides the pseudo inputs that are comparable to the pseudo inputsgenerated by the primary controlled outputs. With such an architectureand pseudo inputs as key link, both MIMO primary control and MIMOconstraint control lead to simple deterministic SISO loop design.

Mechanisms for obtaining a simple, deterministic, robust and systematicsolution with explicit physical meaning for the advanced multi-variablecontrol with multiple constraint problems include: (1) To classifyconstraint candidates into specific subsets, and each constraint subsetis corresponding to one trade-off target of primary control outputs; (2)The number of constraint subsets should be equal to or less than theprimary control handles; (3) For high dimension multiple constraints,i.e., constraints from different subsets to be active at same time, theyshould be decoupled before constructing the SISO constraint controllersin each subset; (4) Each constraint subset is calculating its trade-offtarget—the specific pseudo input based on each of the constraintregulators in this constraint subset; (5) MIMO primary control shoulddecouple the primary controlled outputs; (6) Therefore, multi-dimensionconstraints should be decoupled one dimension from another dimension anddecoupled from the non-traded off primary controlled outputs—it followsthat the above constraint controller is a decoupled SISO regulator withdesired dynamics based on constraint MIMO dynamics inversion withrespect to its relative degree; (7) The most limiting constraint shouldbe resulted from comparing the pseudo inputs generated by theconstraints in each subset and the associated primary controlled outputbased on pre-determined selection logic; (8) The pseudo input generatedby the most limiting constraint controller is applied to replace theprimary control that is pre-determined to trade off; (9) The mostlimiting constraint(s) active/inactive transition is managed by theselection logic smoothly (Example is as shown in FIG. 6).

The design procedure and approaches of the Advanced Multiple VariableControl with High Dimension Multiple Constraints is described below.

The Original Controlled Plant

Without loss of generality, assume the original controlled plant is:

x _(k+1) =f(x _(k) ,{dot over (u)} _(k) ,d _(k))

y _(k) =h(x _(k) ,{dot over (u)} _(k) ,d _(k))

At sample k, the system states x_(k), the inputs {dot over (u)}_(k−1),and the disturbances d_(k) are known. Thus, the deviation variables areexpressed about this current operating condition, i.e. x_(k), {dot over(u)}_(k−1), d_(k), y_(k) ⁻=h(x_(k),{dot over (u)}_(k−1),d_(k)).

Define the deviation variables from these conditions,

{tilde over (x)} _(j) =x _(j) −x _(k)

{tilde over ({dot over (u)}={dot over (u)} _(j) −{dot over (u)} _(k−1)

{tilde over (d)} _(j) =d _(j) −d _(k)

{tilde over (y)} _(j) =y _(j) −y _(k) ⁻

The local linearized model of the system in terms of deviation variablesmay be derived

$\begin{matrix}{{x_{k + 1} - x_{k}} = {\overset{\sim}{x}}_{k + 1}} \\{= \left. {{f\left( {x_{k},u_{k - 1},d_{k}} \right)} - x_{k} + \frac{\partial f}{\partial x}} \middle| {}_{k,{k - 1}}{\left( {x_{k} - x_{k}} \right) + \frac{\partial f}{\partial\overset{.}{u}}} \right|_{k,{k - 1}}} \\{\left. {\left( {{\overset{.}{u}}_{k} - {\overset{.}{u}}_{k - 1}} \right) + \frac{\partial f}{\partial d}} \middle| {}_{k,{k - 1}}\left( {d_{k} - d_{k}} \right) \right.} \\{= {F_{k} + {A\; {\overset{\sim}{x}}_{k}} + {B\; {\overset{.}{u}}_{k}} + {B_{d}{\overset{\sim}{d}}_{k}}}}\end{matrix}$

Approximate F_(k)≈{circumflex over (x)}_(k)=x_(k)−x_(k−1), and it istreated as a known initial condition for {tilde over (x)}_(k+1) atsample k, or, autonomous response of the system states over one controlsample free from any control action update, i.e. {tilde over ({dot over(u)}_(k)=0.

$\begin{matrix}{{y_{k} - y_{k}^{-}} = {\overset{\sim}{y}}_{k}} \\{= \left. {{h\left( {x_{k},u_{k - 1},d_{k}} \right)} - y_{k}^{-} + \frac{\partial h}{\partial x}} \middle| {}_{k,{k - 1}}{\left( {x_{k} - x_{k}} \right) + \frac{\partial h}{\partial\overset{.}{u}}} \right|_{k,{k - 1}}} \\{\left. {\left( {{\overset{.}{u}}_{k} - {\overset{.}{u}}_{k - 1}} \right) + \frac{\partial h}{\partial d}} \middle| {}_{k,{k - 1}}\left( {d_{k} - d_{k}} \right) \right.} \\{= {{C{\overset{\sim}{x}}_{k}} + {D_{u}\; {\overset{\sim}{\overset{.}{u}}}_{k}} + {D_{d}{\overset{\sim}{d}}_{k}}}}\end{matrix}$

The generic perturbation model based on plant dynamics partials ispresented below (for example, it can be based on engine dynamicspartials from cycle study):

{tilde over (x)}(k+1)=A{tilde over (x)}(k)+B{tilde over ({dot over(u)}(k)+B _(d) {tilde over (d)}(k)+F _(k)

{tilde over (y)}(k)=C{tilde over (x)}(k)+D _(d) {tilde over (d)}(k)

{tilde over (y)} _(c)(k)=C _(c) {tilde over (x)}(k)+D _(cd) {tilde over(d)}(k)

Where {tilde over (x)}εR^(n×1), {tilde over ({dot over (u)}εR^(m×1),{tilde over (y)}εR^(m×1), {tilde over (d)}εR^(q×1), {tilde over(y)}_(c)εR^(p×1), p>m.

Approximations:

d(k)−d(k−1)=d(k+1)−d(k), . . .

{tilde over (d)}(k+1)=d(k+1)−d(k)≈{circumflex over (d)}(k),

Without loss of generality and for clear formulation of the designprocess, assume that the primary control has 3 control inputs and 3outputs, i.e. 3×3 control,

${\overset{.}{u} = \begin{bmatrix}{\overset{.}{u}}_{1} \\{\overset{.}{u}}_{2} \\{\overset{.}{u}}_{3}\end{bmatrix}},{y = \begin{bmatrix}y_{1} \\y_{2} \\y_{3}\end{bmatrix}},{y_{c} = {\begin{bmatrix}y_{c\; 1} \\y_{c\; 2} \\y_{c\; 3} \\y_{c\; 4}\end{bmatrix}.}}$

And assume that y₁ has relative degree 3, and y₂ and y₃ both haverelative degree 2; and y_(c1) has relative degree 3, and y_(c2), y_(c3)and y_(c4) both have relative degree 2.

The Primary Control Based on Original Controlled Plant

Use relative degree concept and dynamics inversion approach, the primarycontrol output response is derived below.

Assume the relative degree of {tilde over (y)}_(i) to {tilde over ({dotover (u)} is Rd_(i)>1, then the primary controlled output response are:

$\begin{matrix}{{{\overset{\sim}{y}}_{i}\left( {k + 1} \right)} = {{C_{i}A{\overset{\sim}{x}(k)}} + {C_{i}B{\overset{\sim}{\overset{.}{u}}(k)}} + {C_{i}B_{d}{\overset{\sim}{d}(k)}} + {C_{i}F_{k}} + {D_{di}{\overset{\sim}{d}\left( {k + 1} \right)}}}} \\{= {{C_{i}F_{k}} + {D_{di}{\hat{d}(k)}}}} \\{{= {{K_{f,i}^{1}F_{k}} + {K_{d,i}^{1}{\hat{d}(k)}}}},}\end{matrix}$ $\begin{matrix}{{{\overset{\sim}{y}}_{i}\left( {k + 2} \right)} = {{C_{i}A^{2}{\overset{\sim}{x}(k)}} + {C_{i}A\; B{\overset{\sim}{\overset{.}{u}}(k)}} + {C_{i}A\; B_{d}{\overset{\sim}{d}(k)}} + {C_{i}B_{d}{\overset{\sim}{d}\left( {k + 1} \right)}} +}} \\{{{{C_{i}\left( {A + I} \right)}F_{k}} + {D_{di}{\overset{\sim}{d}\left( {k + 2} \right)}}}} \\{= {{{C_{i}\left( {A + I} \right)}F_{k}} + {\left( {{C_{i}B_{d}} + {2D_{di}}} \right){\hat{d}(k)}}}} \\{{= {{K_{f,i}^{2}F_{k}} + {K_{d,i}^{2}{\hat{d}(k)}}}},}\end{matrix}$ … $\begin{matrix}{{\overset{\sim}{y_{i}}\left( {k + {Rd}_{i}} \right)} = {{C_{i}A^{{Rd}_{i}}{\overset{\sim}{x}(k)}} + {C_{i}A^{{Rd}_{i} - 1}B{\overset{\sim}{\overset{.}{u}}(k)}} +}} \\{{\begin{pmatrix}{{C_{i}A^{{Rd}_{i} - 1}B_{d}{\overset{\sim}{d}(k)}} + \ldots +} \\{{C_{i}B_{d}{\overset{\sim}{d}\left( {k + {Rd}_{i} - 1} \right)}} + \ldots + {D_{di}{\overset{\sim}{d}\left( {k + {Rd}_{i}} \right)}}}\end{pmatrix} +}} \\{{{C_{i}\left( {A^{{Rd}_{i} - 1} + \ldots + A + I} \right)}F_{k}}} \\{{= {{E_{i}{\overset{\sim}{\overset{.}{u}}(k)}} + {K_{f,i}F_{k}} + {K_{d,i}{\hat{d}(k)}}}},}\end{matrix}$Where  E_(i) = C_(i)A^(Rd_(i) − 1)B, K_(f, i) = C_(i)(A^(Rd_(i) − 1) + … + A + I), K_(d, i) = C_(i)A^(Rd_(i) − 2)B_(d) + … + (Rd_(i) − 1) ⋅ C_(i)B_(d) + Rd_(i) ⋅ D_(di).

The current controlled output response in general is described below:

${\overset{\sim}{y}\left( {k + {Rd}} \right)} = {{E{\overset{\sim}{\overset{.}{u}}(k)}} + {K_{f}F_{k}} + {K_{d}{\hat{d}(k)}}}$Where ${{\overset{\sim}{y}\left( {k + {Rd}} \right)} = \begin{bmatrix}{{\overset{\sim}{y}}_{1}\left( {k + 3} \right)} \\{{\overset{\sim}{y}}_{2}\left( {k + 2} \right)} \\{{\overset{\sim}{y}}_{3}\left( {k + 2} \right)}\end{bmatrix}},{E = {\begin{bmatrix}E_{1} \\E_{2} \\E_{3}\end{bmatrix} = \begin{bmatrix}{C_{1}A^{2}B} \\{C_{2}{AB}} \\{C_{3}{AB}}\end{bmatrix}}},{K_{f} = {\begin{bmatrix}K_{f,1} \\K_{f,2} \\K_{f,3}\end{bmatrix} = \begin{bmatrix}{C_{1}\left( {A^{2} + A + I} \right)} \\{C_{2}\left( {A + I} \right)} \\{C_{3}\left( {A + I} \right)}\end{bmatrix}}},{K_{d} = {\begin{bmatrix}K_{d,1} \\K_{d,2} \\K_{d,3}\end{bmatrix} = {\begin{bmatrix}{{{C_{1}\left( {A + {2I}} \right)}B_{d}} + {3D_{d\; 1}}} \\{{C_{2}B_{d}} + {2D_{d\; 2}}} \\{{C_{3}B_{d}} + {2D_{d\; 2}}}\end{bmatrix}.}}}$

Further, the dynamics of controlled output y(k) is desired to track thereference y_(r)(k), i.e., let

ŷ _(i)(k+j)=y _(ri)(k+j)−y _(i)(k),i=1,2,3;j=0, . . . ,Rd _(i),

the desired control tracking performance is shaped below:

(ŷ ₁(k+3)−{tilde over (y)} ₁(k+3))+k _(1,2)(ŷ ₁(k+2)−{tilde over (y)}₁(k+2))+k _(1,1)(ŷ ₁(k+1)−{tilde over (y)} ₁(k+1))+k _(1,0)(ŷ₁(k)−{tilde over (y)} ₁(k))=0

(ŷ ₂(k+2)−{tilde over (y)} ₂(k+2))+k _(2,1)(ŷ ₂(k+1)−{tilde over (y)}₂(k+1))+k _(2,0)(ŷ ₂(k)−{tilde over (y)} ₂(k))=0

(ŷ ₃(k+2)−{tilde over (y)} ₃(k+2))+k _(3,1)(ŷ ₃(k+1)−{tilde over (y)}₃(k+1))+k _(3,0)(ŷ ₃(k)−{tilde over (y)} ₃(k))=0

Properly choose k_(i,j), i=1, . . . , 3; j=0, . . . , Rd_(i)−1 such thatthe following polynomial

ρ^(Rd) ^(i) + . . . +k _(i,j)ρ^(j) + . . . +k _(i,1) ρ+k _(i,0)=0  (Eq.130)

has its eigenvalues all within the unit circle, then the primary controldynamics is asymptotically stable.

Usually y_(ri) (k) is time-varying, the approximated {tilde over(y)}_(ri) (k+j), j=1, . . . , Rd_(i) can be obtained by usingextrapolation (such as linear format, exponential format, etc.). Let{tilde over (y)}_(ri)(k)=y_(ri)(k)−y_(ri)(k−1). Approximate {tilde over(y)}_(ri)(k+1)≈α_(i){tilde over (y)}_(ri)(k), {tilde over(y)}_(r)(k+2)≈α²{tilde over (y)}_(r)(k), . . . .

The desired controlled output tracking response:

$\begin{matrix}{{{\overset{\sim}{y}}_{1}\left( {k + 3} \right)} = {{{\hat{y}}_{1}\left( {k + 3} \right)} + {k_{1,2}\left( {{{\hat{y}}_{1}\left( {k + 2} \right)} - {{\overset{\sim}{y}}_{1}\left( {k + 2} \right)}} \right)} +}} \\{{{k_{1,1}\left( {{{\hat{y}}_{1}\left( {k + 1} \right)} - {{\overset{\sim}{y}}_{1}\left( {k + 1} \right)}} \right)} + {k_{1,0}\left( {{{\hat{y}}_{1}(k)} - {{\overset{\sim}{y}}_{1}(k)}} \right)}}} \\{= {\left\lbrack {{{\hat{y}}_{1}\left( {k + 3} \right)} + {k_{1,2}{{\hat{y}}_{1}\left( {k + 2} \right)}} + {k_{1,1}{{\hat{y}}_{1}\left( {k + 1} \right)}} + {k_{1,0}{{\hat{y}}_{1}(k)}}} \right\rbrack -}} \\{\left\lbrack {{k_{1,2}{{\overset{\sim}{y}}_{1}\left( {k + 2} \right)}} + {k_{1,1}{{\overset{\sim}{y}}_{1}\left( {k + 1} \right)}}} \right\rbrack} \\{= {\left\lbrack {K_{e\; 1}{{\hat{y}}_{1}^{*}(k)}} \right\rbrack - \begin{bmatrix}{{k_{1,2}\left( {{K_{f,1}^{2}F_{k}} + {K_{d,1}^{2}{\hat{d}(k)}}} \right)} +} \\{k_{1,1}\left( {{K_{f,1}^{1}F_{k}} + {K_{d,1}^{1}{\hat{d}(k)}}} \right)}\end{bmatrix}}} \\{= {{K_{e\; 1}{{\hat{y}}_{1}^{*}(k)}} - {K_{{df},1}F_{k}} - {K_{{dd},1}{\hat{d}(k)}}}}\end{matrix}$ $\begin{matrix}{{{\overset{\sim}{y}}_{2}\left( {k + 2} \right)} = {{{\hat{y}}_{2}\left( {k + 2} \right)} + {k_{2,1}\left( {{{\hat{y}}_{2}\left( {k + 1} \right)} - {{\overset{\sim}{y}}_{2}\left( {k + 1} \right)}} \right)} +}} \\{{k_{2,0}\left( {{{\hat{y}}_{2}(k)} - {{\overset{\sim}{y}}_{2}(k)}} \right)}} \\{= {\left\lbrack {{{\hat{y}}_{2}\left( {k + 2} \right)} + {k_{2,1}{{\hat{y}}_{2}\left( {k + 1} \right)}} + {k_{2,0}{{\hat{y}}_{2}(k)}}} \right\rbrack -}} \\{\left\lbrack {k_{2,1}{{\overset{\sim}{y}}_{2}\left( {k + 1} \right)}} \right\rbrack} \\{= {\left\lbrack {K_{e\; 2}{{\hat{y}}_{2}^{*}(k)}} \right\rbrack - \left\lbrack {k_{2,1}\left( {{K_{f,2}^{1}F_{k}} + {K_{d,2}^{1}{\hat{d}(k)}}} \right)} \right\rbrack}} \\{= {{K_{e\; 2}{{\hat{y}}_{2}^{*}(k)}} - {K_{{df},2}F_{k}} - {K_{{dd},2}{\hat{d}(k)}}}}\end{matrix}$ $\begin{matrix}{{{\overset{\sim}{y}}_{3}\left( {k + 2} \right)} = {{{\hat{y}}_{3}\left( {k + 2} \right)} + {k_{3,1}\left( {{{\hat{y}}_{3}\left( {k + 1} \right)} - {{\overset{\sim}{y}}_{3}\left( {k + 1} \right)}} \right)} +}} \\{{k_{3,0}\left( {{{\hat{y}}_{3}(k)} - {{\overset{\sim}{y}}_{3}(k)}} \right)}} \\{= {\left\lbrack {{{\hat{y}}_{3}\left( {k + 2} \right)} + {k_{3,1}{{\hat{y}}_{3}\left( {k + 1} \right)}} + {k_{3,0}{{\hat{y}}_{3}(k)}}} \right\rbrack -}} \\{\left\lbrack {k_{3,1}{{\overset{\sim}{y}}_{3}\left( {k + 1} \right)}} \right\rbrack} \\{= {\left\lbrack {K_{e\; 3}{{\hat{y}}_{3}^{*}(k)}} \right\rbrack - \left\lbrack {k_{3,1}\left( {{K_{f,3}^{1}F_{k}} + {K_{d,3}^{1}{\hat{d}(k)}}} \right)} \right\rbrack}} \\{= {{K_{e\; 3}{{\hat{y}}_{3}^{*}(k)}} - {K_{{df},3}F_{k}} - {K_{{dd},3}{\hat{d}(k)}}}}\end{matrix}$ Where ${K_{e\; 1} = \begin{bmatrix}1 & k_{1,2} & k_{1,1} & k_{1,0}\end{bmatrix}},{{{\hat{y}}_{1}^{*}(k)} = {\begin{bmatrix}{{y_{r\; 1}(k)} - {y_{1}(k)}} \\{{y_{r\; 1}(k)} - {y_{1}(k)}} \\{{y_{r\; 1}(k)} - {y_{1}(k)}} \\{{y_{r\; 1}(k)} - {y_{1}(k)}}\end{bmatrix} + \begin{bmatrix}{\alpha_{1}^{3}{{\overset{\sim}{y}}_{r\; 1}(k)}} \\{\alpha_{1}^{2}{{\overset{\sim}{y}}_{r\; 1}(k)}} \\{\alpha_{1}{{\overset{\sim}{y}}_{r\; 1}(k)}} \\0\end{bmatrix}}},{K_{{df},1} = \left( {{k_{1,2}K_{f,1}^{2}} + {k_{1,1}K_{f,1}^{1}}} \right)},{{K_{{dd},1} = \left( {{k_{1,2}K_{d,1}^{2}} + {k_{1,1}K_{d,1}^{1}}} \right)};}$${K_{e\; 2} = \begin{bmatrix}1 & k_{2,1} & k_{2,0}\end{bmatrix}},{{{\hat{y}}_{2}^{*}(k)} = {\begin{bmatrix}{{y_{r\; 2}(k)} - {y_{2}(k)}} \\{{y_{r\; 2}(k)} - {y_{2}(k)}} \\{{y_{r\; 2}(k)} - {y_{2}(k)}}\end{bmatrix} + \begin{bmatrix}{\alpha_{2}^{2}{{\overset{\sim}{y}}_{r\; 2}(k)}} \\{\alpha_{2}{{\overset{\sim}{y}}_{r\; 2}(k)}} \\0\end{bmatrix}}},{K_{{df},2} = {k_{2,1}K_{f,2}^{1}}},{{K_{{dd},2} = {k_{2,1}K_{d,2}^{1}}};}$${K_{e\; 3} = \begin{bmatrix}1 & k_{3,1} & k_{3,0}\end{bmatrix}},{{{\hat{y}}_{3}^{*}(k)} = {\begin{bmatrix}{{y_{r\; 3}(k)} - {y_{3}(k)}} \\{{y_{r\; 3}(k)} - {y_{3}(k)}} \\{{y_{r\; 3}(k)} - {y_{3}(k)}}\end{bmatrix} + \begin{bmatrix}{\alpha_{3}^{2}{{\overset{\sim}{y}}_{r\; 3}(k)}} \\{\alpha_{3}{{\overset{\sim}{y}}_{r\; 3}(k)}} \\0\end{bmatrix}}},{K_{{df},3} = {k_{3,1}K_{f,3}^{1}}},{{K_{{dd},3} = {k_{3,1}K_{d,3}^{1}}};}$

If y_(ri)(k) is constant,

${{{\hat{y}}_{1}^{*}(k)} = \begin{bmatrix}{{y_{r\; 1}(k)} - {y_{1}(k)}} \\{{y_{r\; 1}(k)} - {y_{1}(k)}} \\{{y_{r\; 1}(k)} - {y_{1}(k)}} \\{{y_{r\; 1}(k)} - {y_{1}(k)}}\end{bmatrix}},{{{\hat{y}}_{2}^{*}(k)} = \begin{bmatrix}{{y_{r\; 2}(k)} - {y_{2}(k)}} \\{{y_{r\; 2}(k)} - {y_{2}(k)}} \\{{y_{r\; 2}(k)} - {y_{2}(k)}}\end{bmatrix}},{{{\hat{y}}_{3}^{*}(k)} = {\begin{bmatrix}{{y_{r\; 3}(k)} - {y_{3}(k)}} \\{{y_{r\; 3}(k)} - {y_{3}(k)}} \\{{y_{r\; 3}(k)} - {y_{3}(k)}}\end{bmatrix}.}}$

Note that the free response y_(i) ^(f) is not dependent on ũ(k), onlydepend on F_(k) and {circumflex over (d)}(k). Further the desiredcontrolled output response in a compact way,

$\left. {{\overset{\sim}{y}\left( {k + {Rd}} \right)} = {{K_{RE}{{\hat{y}}^{*}(k)}} - {K_{df}F_{k}} - {K_{dd}{\hat{d}(k)}}}} \right)$Where ${K_{RE} = \begin{bmatrix}K_{e\; 1} & 0 & 0 \\0 & K_{e\; 2} & 0 \\0 & 0 & K_{e\; 3}\end{bmatrix}},{{{\hat{y}}^{*}(k)} = \begin{bmatrix}{{\hat{y}}_{1}^{*}(k)} \\{{\hat{y}}_{2}^{*}(k)} \\{{\hat{y}}_{3}^{*}(k)}\end{bmatrix}},{K_{df} = \begin{bmatrix}K_{{df},1} \\K_{{df},2} \\K_{{df},3}\end{bmatrix}},{K_{dd} = \begin{bmatrix}K_{{dd},1} \\K_{{dd},2} \\K_{{dd},3}\end{bmatrix}}$

Define the pseudo input as:

{dot over (v)} _(i)(k)=K _(RE)(i,i)ŷ* _(i)(k)

Compare the above desired controlled output response with the currentcontrolled output response, the primary decoupling control based ondynamics inversion is obtained below:

E{tilde over ({dot over (u)}={dot over (v)}(k)−(K _(df) +K _(f))F_(k)−(K _(dd) +K _(d)){circumflex over (d)}(k)

ũ(k)=K _(V) {dot over (v)}(k)+K _(F) F _(k) +K _(D) {circumflex over(d)}(k)

The resulting control decouples the SISO loop {dot over (v)}_(i)→y_(i)from SISO loop {dot over (v)}_(j)→y_(j), j≠i, therefore each output istracking its own reference, i.e., being controlled by its own referenceonly.

The New Controlled Plant Based on Primary Control for Constraint Control

Substitute the primary decoupling control law into the originalcontrolled plant, yields the decoupled new controlled plant:

{tilde over (x)}(k+1)=A{tilde over (x)}(k)+B _(d) {tilde over (d)}(k)+BK_(V) {dot over (v)}(k)+(BK _(F) +I)F _(k) +BK _(D) {circumflex over(d)}(k)

{tilde over (y)}(k)=C{tilde over (x)}(k)=D _(d) {tilde over (d)}(k)

{tilde over (y)} _(c)(k)=C _(c) {tilde over (x)}(k)+D _(cd) {tilde over(d)}(k)

ũ(k)=K _(V) {dot over (v)}(k)+K _(F) F _(k) +K _(D) {circumflex over(d)}(k)

When the same control handle ũ(k) is needed to enforce certain selectedactive constraint(s) while remaining the primary output trackingimpacted least, the certain one(s) of primary output tracking will betraded off by switching to the selected most limiting control modeinstead of allowing its reference to be altered via adding Δy_(r)(k) toy_(r)(k). For the new controlled plant, the control input is {dot over(v)}.

The Constraint Decoupling Control Based on the New Controlled Plant

The new controlled plant is:

{tilde over (x)}(k+1)=A{tilde over (x)}(k)+F _(k) ^(c) +B _(v) {dot over(v)}(k)+B _(d) ^(c) {circumflex over (d)}(k)+B _(d) {tilde over (d)}(k)

{tilde over (y)} _(c)(k)=C _(c) {tilde over (x)}(k)+D _(cd) {tilde over(d)}(k)

Current constraint output responses are, respectively,

$\begin{matrix}{{{\overset{\sim}{y}}_{ci}\left( {k + 1} \right)} = {{C_{ci}A{\overset{\sim}{x}(k)}} + {C_{ci}B_{v}{\overset{.}{v}(k)}} + {C_{ci}F_{k}^{c}} + {C_{i}\left( {{B_{d}^{c}{\hat{d}(k)}} + {B_{d}{\overset{\sim}{d}(k)}}} \right)} +}} \\{{D_{cdi}{\overset{\sim}{d}\left( {k + 1} \right)}}} \\{= {{C_{ci}F_{k}^{c}} + {\left( {{C_{i}B_{d}^{c}} + D_{cdi}} \right){\hat{d}(k)}}}} \\{= {{K_{{cfp},i}^{1}F_{k}^{c}} + {K_{{cd},i}^{1}{\hat{d}(k)}}}} \\{{= {{K_{{cf},i}^{1}F_{k}} + {K_{{cd},i}^{1}{\hat{d}(k)}}}},}\end{matrix}$$\mspace{20mu} {{{{Since}\mspace{14mu} F_{k}^{c}} = {\left( {I + {B_{u}K_{F}}} \right)F_{k}}},\mspace{20mu} {{K_{{cfp},i}^{1}\left( {I + {B_{u}K_{F}}} \right)} = {K_{{cf},i}^{1}.\begin{matrix}{{{\overset{\sim}{y}}_{ci}\left( {k + 2} \right)} = {{C_{ci}A^{2}{\overset{\sim}{x}(k)}} + {C_{ci}A\; B_{v}{\overset{.}{v}(k)}} + {{C_{ci}\left( {A + I} \right)}F_{k}^{c}} +}} \\{{{C_{ci}{A\left( {{B_{d}^{c}{\hat{d}(k)}} + {B_{d}{\overset{\sim}{d}(k)}}} \right)}} + \ldots +}} \\{{{C_{ci}\left( {{B_{d}^{c}{\hat{d}\left( {k + 1} \right)}} + {B_{d}{\overset{\sim}{d}\left( {k + 1} \right)}}} \right)} +}} \\{{D_{cdi}{\overset{\sim}{d}\left( {k + 2} \right)}}} \\{= {{{C_{ci}\left( {A + I} \right)}F_{k}^{c}} + {\left( {{C_{ci}A\; B_{d}^{c}} + {C_{ci}B_{d}^{c}} + {C_{ci}B_{d}} + {2D_{di}}} \right){\hat{d}(k)}}}} \\{= {{K_{{cfp},i}^{2}F_{k}^{c}} + {K_{{cd},i}^{2}{\hat{d}(k)}}}} \\{= {{K_{{cf},i}^{2}F_{k}^{c}} + {K_{{cd},i}^{2}{\hat{d}(k)}}}}\end{matrix}}}}$   … $\begin{matrix}{{{\overset{\sim}{y}}_{ci}\left( {k + 3} \right)} = {{C_{ci}A^{3}{\overset{\sim}{x}(k)}} + {C_{ci}A^{2}\; B_{v}{\overset{.}{v}(k)}} + {{C_{ci}\left( {A^{2} + A + I} \right)}\; F_{k}^{c}} + \ldots +}} \\{\begin{bmatrix}\begin{matrix}{{C_{ci}{A^{2}\left( {{B_{d}^{c}{\hat{d}(k)}} + {B_{d}{\overset{\sim}{d}(k)}}} \right)}} +} \\{{C_{i}{A\left( {{B_{d}^{c}{\hat{d}\left( {k + 1} \right)}} + {B_{d}{\overset{\sim}{d}\left( {k + 1} \right)}}} \right)}} +}\end{matrix} \\{{C_{i\;}\left( {{B_{d}^{c}{\hat{d}\left( {k + 2}\; \right)}} + {B_{d}{\overset{\sim}{d}\left( {k + 2} \right)}}} \right)} + {D_{cdi}{\overset{\sim}{d}\left( {k + 3} \right)}}}\end{bmatrix}} \\{= {{C_{ci}A^{2}B_{v}{\overset{.}{v}(k)}} + {{C_{ci}\left( {A^{2} + A + I} \right)}F_{k}^{c}} +}} \\{= {\left( {{C_{ci}A^{2}B_{d}^{c}} + {C_{ci}{A\left( {B_{d}^{c} + B_{d}} \right)}} + {2{C_{ci}\left( {B_{d}^{c} + B_{d}} \right)}} + {3D_{cdi}}} \right){\hat{d}(k)}}} \\{= {{E_{ci}{\overset{.}{v}(k)}} + {K_{{cfp},i}F_{k}^{c}} + {K_{{cd},i}{\hat{d}(k)}}}} \\{{= {{E_{ci}{\overset{.}{v}(k)}} + {K_{{cf},i}F_{k}} + {K_{{cd},i}{\hat{d}(k)}}}},}\end{matrix}$$\mspace{20mu} {{{{Where}\mspace{14mu} E_{ci}} = {C_{ci}A^{{Rd}_{i} - 1}B_{v}}},{K_{{cfp},i} = {C_{ci}\left( {A^{{Rd}_{i} - 1} + \ldots + A + I} \right)}},{K_{{{cd},i}\;} = {{C_{ci}A^{{Rd}_{i} - 1}B_{d}^{c}} + {C_{ci}{A^{{Rd}_{i} - 2}\left( {B_{d}^{c} + B_{d}} \right)}} + \ldots + {\left( {{Rd}_{i} - 1} \right) \cdot {C_{ci}\left( {B_{d}^{c} + B_{d}} \right)}} + {{Rd}_{i} \cdot {D_{cdi}.\mspace{20mu} \begin{matrix}{\begin{bmatrix}{{\overset{\sim}{y}}_{c\; 1}\left( {k + 3} \right)} \\{{\overset{\sim}{y}}_{c\; 2}\left( {k + 2} \right)} \\{{\overset{\sim}{y}}_{c\; 3}\left( {k + 2} \right)} \\{{\overset{\sim}{y}}_{c\; 4}\left( {k + 2} \right)}\end{bmatrix} = {{\begin{bmatrix}{C_{c\; 1}A^{2}B_{v}} \\{C_{c\; 2}A\; B_{v}} \\{C_{c\; 3}A\; B_{v}} \\{C_{c\; 4}A\; B_{v}}\end{bmatrix}{\overset{.}{v}(k)}} + {\begin{bmatrix}{C_{c\; 1}\left( {A^{2} + A + I} \right)} \\{C_{c\; 2}\left( {A + I} \right)} \\{C_{c\; 3}\left( {A + I} \right)} \\{C_{c\; 4}\left( {A + I} \right)}\end{bmatrix}F_{k}^{c}} +}} \\{{\begin{bmatrix}{{C_{c\; 1}\left( {{A^{2}B_{d}^{c}} + {\left( {A + {2\; I}} \right)\left( {B_{d}^{c} + B_{d}} \right)}} \right)} + {3D_{{cd}\; 1}}} \\{{C_{c\; 2}\left( {{A\; B_{d}^{c}} + \left( {B_{d}^{c} + B_{d}} \right)} \right)} + {2D_{{cd}\; 2}}} \\{{C_{c\; 3}\left( {{A\; B_{d}^{c}} + \left( {B_{d}^{c} + B_{d}} \right)} \right)} + {2D_{{cd}\; 3}}} \\{{C_{c\; 4}\left( {{A\; B_{d}^{c}} + \left( {B_{d}^{c} + B_{d}} \right)} \right)} + {2D_{{cd}\; 4}}}\end{bmatrix}{\hat{d}(k)}}} \\{= {{E_{c}{\overset{.}{v}(k)}} + {{K_{cfp}\left( {I + {B_{u}K_{F}}} \right)}F_{k}} + {K_{cd}{\hat{d}(k)}}}} \\{= {{E_{c}{\overset{.}{v}(k)}} + {K_{cf}F_{k}} + {K_{cd}{\hat{d}(k)}}}}\end{matrix}}}}}}$

In general, the desired constraint response is to assure the trackingerror and its derivatives (up to the constraint's relative degree) go tozero, let

ŷ _(ci)(k+j)=y _(rci)(k+j)−y _(ci)(k),i=1, . . . ,4;j=0,1, . . . ,Rd_(ci),

ŷ _(c1)(k+3)−{tilde over (y)} _(c1)(k+3))+k _(c1,2)(ŷ _(c1)(k+2)−{tildeover (y)} _(c1)(k+2))+k _(c1,1)(ŷ _(c1)(k+1)−{tilde over (y)}_(c1)(k+1))+k _(1,0)(ŷ _(c1)(k)−{tilde over (y)} _(c1)(k))

ŷ _(c2)(k+2)−{tilde over (y)} _(c2)(k+2))+k _(c2,1)(ŷ _(c2)(k+1)−{tildeover (y)} _(c2)(k+1))+k _(c2,0)(ŷ _(c2)(k)−{tilde over (y)} _(c2)(k))=0

ŷ _(c3)(k+2)−{tilde over (y)} _(c3)(k+2))+k _(c3,1)(ŷ _(c3)(k+1)−{tildeover (y)} _(c3)(k+1))+k _(c3,0)(ŷ _(c3)(k)−{tilde over (y)} _(c3)(k))=0

ŷ _(c4)(k+2)−{tilde over (y)} _(c4)(k+2))+k _(c4,1)(ŷ _(c4)(k+1)−{tildeover (y)} _(c4)(k+1))+k _(c4,0)(ŷ _(c4)(k)−{tilde over (y)} _(c4)(k))=0

Properly choose the above coefficients k_(ci,j), i=1, . . . ,4; j=0, . .. , Rd_(ci)−1 such that the eigenvalues of the following polynomial

ρ^(Rd) ^(ci) + . . . +k _(ci,j)ρ^(j) + . . . +k _(ci,1) ρ+k _(ci,0)=0

are all within the unit circle, then the constraint output trackingdynamics is asymptotically stable.

The desired constraint output tracking response:

$\begin{matrix}{{{\overset{\sim}{y}}_{c\; 1}\left( {k + 3} \right)} = {{{\hat{y}}_{c\; 1}\left( {k + 3} \right)} + {k_{{c\; 1},2}\left( {{{\hat{y}}_{c\; 1}\left( {k + 2} \right)} - {{\overset{\sim}{y}}_{c\; 1}\left( {k + 2} \right)}} \right)} +}} \\{{{k_{{c\; 1},1}\left( {{{\hat{y}}_{c\; 1}\left( {k + 1} \right)} - {{\overset{\sim}{y}}_{c\; 1}\left( {k + 1} \right)}} \right)} + {k_{{c\; 1},0}\left( {{{\hat{y}}_{c\; 1}(k)} - {{\overset{\sim}{y}}_{c\; 1}(k)}} \right)}}} \\{= {\begin{bmatrix}{{{\hat{y}}_{c\; 1}\left( {k + 3} \right)} + {k_{{c\; 1},2}{\hat{y}}_{c\; 1}\left( {k + 2} \right)} +} \\{{k_{{c\; 1},1}{{\hat{y}}_{c\; 1}\left( {k + 1} \right)}} + {k_{{c\; 1},0}{{\hat{y}}_{c\; 1}(k)}}}\end{bmatrix} -}} \\{\left\lbrack {{k_{{c\; 1},2}{{\overset{\sim}{y}}_{c\; 1}\left( {k + 2} \right)}} + {k_{{c\; 1},1}{{\overset{\sim}{y}}_{c\; 1}\left( {k + 1} \right)}}} \right\rbrack} \\{= {\left\lbrack {K_{{ce}\; 1}{{\hat{y}}_{c\; 1}^{*}(k)}} \right\rbrack - \begin{bmatrix}{{k_{{c\; 1},2}\left( {{K_{{cf},1}^{2}F_{k}} + {K_{{cd},1}^{2}{\hat{d}(k)}}} \right)} +} \\{k_{{c\; 1},1}\left( {{K_{{cf},1}^{1}F_{k}} + {K_{{cd},1}^{1}{\hat{d}(k)}}} \right)}\end{bmatrix}}} \\{= {{K_{{ce}\; 1}{{\hat{y}}_{1}^{*}(k)}} - {K_{{cdf},1}F_{k}} - {K_{{cdd},1}{\hat{d}(k)}}}}\end{matrix}$ $\begin{matrix}{{{\overset{\sim}{y}}_{c\; 2}\left( {k + 2} \right)} = {{{\hat{y}}_{c\; 2}\left( {k + 2} \right)} + {k_{{c\; 2},1}\left( {{{\hat{y}}_{c\; 2}\left( {k + 1} \right)} - {{\overset{\sim}{y}}_{c\; 2}\left( {k + 1} \right)}} \right)} +}} \\{{k_{{c\; 2},0}\left( {{{\hat{y}}_{c\; 2}(k)} - {{\overset{\sim}{y}}_{c\; 2}(k)}} \right)}} \\{= {\left\lbrack {{{\hat{y}}_{c\; 2}\left( {k + 2} \right)} + {k_{{c\; 2},1}{{\hat{y}}_{c\; 2}\left( {k + 1} \right)}} + {k_{{c\; 2},0}{{\hat{y}}_{c\; 2}(k)}}} \right\rbrack -}} \\{\left\lbrack {k_{{c\; 2},1}{{\overset{\sim}{y}}_{c\; 2}\left( {k + 1} \right)}} \right\rbrack} \\{= {\left\lbrack {K_{c\; e\; 2}{{\hat{y}}_{c\; 2}^{*}(k)}} \right\rbrack - \left\lbrack {k_{{c\; 2},1}\left( {{K_{{c\; f},2}^{1}F_{k}} + {K_{{c\; d},2}^{1}{\hat{d}(k)}}} \right)} \right\rbrack}} \\{= {{K_{c\; e\; 2}{{\hat{y}}_{c\; 2}^{*}(k)}} - {K_{{cdf},2}F_{k}} - {K_{{c\; {dd}},2}{\hat{d}(k)}}}}\end{matrix}$ $\begin{matrix}{{{\overset{\sim}{y}}_{c\; 3}\left( {k + 2} \right)} = {{{\hat{y}}_{c\; 3}\left( {k + 2} \right)} + {k_{{c\; 3},1}\left( {{{\hat{y}}_{c\; 3}\left( {k + 1} \right)} - {{\overset{\sim}{y}}_{c\; 3}\left( {k + 1} \right)}} \right)} +}} \\{{k_{{c\; 3},0}\left( {{{\hat{y}}_{c\; 3}(k)} - {{\overset{\sim}{y}}_{c\; 3}(k)}} \right)}} \\{= {\left\lbrack {{{\hat{y}}_{c\; 3}\left( {k + 2} \right)} + {k_{{c\; 3},1}{{\hat{y}}_{c\; 3}\left( {k + 1} \right)}} + {k_{{c\; 3},0}{{\hat{y}}_{c\; 3}(k)}}} \right\rbrack -}} \\{\left\lbrack {k_{{c\; 3},1}{{\overset{\sim}{y}}_{c\; 3}\left( {k + 1} \right)}} \right\rbrack} \\{= {\left\lbrack {K_{c\; e\; 3}{{\hat{y}}_{c\; 3}^{*}(k)}} \right\rbrack - \left\lbrack {k_{{c\; 3},1}\left( {{K_{{c\; f},3}^{1}F_{k}} + {K_{{c\; d},3}^{1}{\hat{d}(k)}}} \right)} \right\rbrack}} \\{= {{K_{c\; e\; 3}{{\hat{y}}_{c\; 3}^{*}(k)}} - {K_{{c\; {df}},3}F_{k}} - {K_{{c\; {dd}},3}{\hat{d}(k)}}}}\end{matrix}$ $\begin{matrix}{{{\overset{\sim}{y}}_{c\; 4}\left( {k + 2} \right)} = {{{\hat{y}}_{c\; 4}\left( {k + 2} \right)} + {k_{{c\; 4},1}\left( {{{\hat{y}}_{c\; 4}\left( {k + 1} \right)} - {{\overset{\sim}{y}}_{c\; 4}\left( {k + 1} \right)}} \right)} +}} \\{{k_{{c\; 4},0}\left( {{{\hat{y}}_{c\; 4}(k)} - {{\overset{\sim}{y}}_{c\; 4}(k)}} \right)}} \\{= {\left\lbrack {{{\hat{y}}_{c\; 4}\left( {k + 2} \right)} + {k_{{c\; 4},1}{{\hat{y}}_{c\; 4}\left( {k + 1} \right)}} + {k_{{c\; 4},0}{{\hat{y}}_{c\; 4}(k)}}} \right\rbrack -}} \\{\left\lbrack {k_{{c\; 4},1}{{\overset{\sim}{y}}_{c\; 4}\left( {k + 1} \right)}} \right\rbrack} \\{= {\left\lbrack {K_{c\; e\; 4}{{\hat{y}}_{c\; 4}^{*}(k)}} \right\rbrack - \left\lbrack {k_{{c\; 4},1}\left( {{K_{{c\; f},4}^{1}F_{k}} + {K_{{c\; d},4}^{1}{\hat{d}(k)}}} \right)} \right\rbrack}} \\{= {{K_{c\; e\; 4}{{\hat{y}}_{c\; 4}^{*}(k)}} - {K_{{c\; {df}},4}F_{k}} - {K_{{c\; {dd}},4}{\hat{d}(k)}}}}\end{matrix}$

Where consider constraint references as constant,

${K_{{ce}\; 1} = \left\lfloor \begin{matrix}1 & k_{{c\; 1},2} & k_{{c\; 1},1} & k_{{c\; 1},0}\end{matrix} \right\rfloor},{{{\hat{y}}_{c\; 1}^{*}(k)} = \begin{bmatrix}{{y_{{rc}\; 1}(k)} - {y_{c\; 1}(k)}} \\{{y_{{rc}\; 1}(k)} - {y_{c\; 1}(k)}} \\{{y_{{rc}\; 1}(k)} - {y_{c\; 1}(k)}} \\{{y_{{rc}\; 1}(k)} - {y_{c\; 1}(k)}}\end{bmatrix}},{K_{{cdf},1} = \left( {{k_{{c\; 1},2}K_{{cf},1}^{2}} + {k_{{c\; 1},1}K_{{cf},1}^{1}}} \right)},{{K_{{cdd},1} = \left( {{k_{{c\; 1},2}K_{{cd},1}^{2}} + {k_{{c\; 1},1}K_{{cd},1}^{1}}} \right)};}$${K_{{ce}\; 2} = \left\lfloor \begin{matrix}1 & k_{{c\; 2},1} & k_{{c\; 2},0}\end{matrix} \right\rfloor},{{{\hat{y}}_{c\; 2}^{*}(k)} = \begin{bmatrix}{{y_{{rc}\; 2}(k)} - {y_{c\; 2}(k)}} \\{{y_{{rc}\; 2}(k)} - {y_{c\; 2}(k)}} \\{{y_{{rc}\; 2}(k)} - {y_{c\; 2}(k)}}\end{bmatrix}},{K_{{cdf},2} = {k_{{c\; 2},1}K_{{cf},2}^{1}}},{{K_{{cdd},2} = {k_{{c\; 2},1}K_{{cd},2}^{1}}};}$${K_{{ce}\; 3} = \left\lfloor \begin{matrix}1 & k_{{c\; 3},1} & k_{{c\; 3},0}\end{matrix} \right\rfloor},{{{\hat{y}}_{c\; 3}^{*}(k)} = \begin{bmatrix}{{y_{{rc}\; 3}(k)} - {y_{c\; 3}(k)}} \\{{y_{{rc}\; 3}(k)} - {y_{c\; 3}(k)}} \\{{y_{{rc}\; 3}(k)} - {y_{c\; 3}(k)}}\end{bmatrix}},{K_{{cdf},3} = {k_{{c\; 3},1}K_{{cf},3}^{1}}},{{K_{{cdd},3} = {k_{{c\; 3},1}K_{{cd},3}^{1}}};}$${K_{{ce}\; 4} = \left\lfloor \begin{matrix}1 & k_{{c\; 4},1} & k_{{c\; 4},0}\end{matrix} \right\rfloor},{{{\hat{y}}_{c\; 4}^{*}(k)} = \begin{bmatrix}{{y_{{rc}\; 4}(k)} - {y_{c\; 4}(k)}} \\{{y_{{rc}\; 4}(k)} - {y_{c\; 4}(k)}} \\{{y_{{rc}\; 4}(k)} - {y_{c\; 4}(k)}}\end{bmatrix}},{K_{{cdf},4} = {k_{{c\; 4},1}K_{{cf},4}^{1}}},{K_{{cdd},4} = {k_{{c\; 4},1}{K_{{cd},4}^{1}.}}}$

Further the desired constraint output response in a compact way,

${{\overset{\sim}{y}}_{c}\left( {k + {Rd}_{c}} \right)} = {{K_{cRE}{{\hat{y}}_{c}^{*}(k)}} - {K_{cdf}F_{k}} - {K_{cdd}{\hat{d}(k)}}}$Where ${K_{cRE} = \begin{bmatrix}K_{{ce}\; 1} & 0 & 0 & 0 \\0 & K_{{ce}\; 2} & 0 & 0 \\0 & 0 & K_{{ce}\; 3} & 0 \\0 & 0 & 0 & K_{{ce}\; 4}\end{bmatrix}},{{{\hat{y}}_{c}^{*}(k)} = \begin{bmatrix}{{\hat{y}}_{c\; 1}^{*}(k)} \\{{\hat{y}}_{c\; 2}^{*}(k)} \\{{\hat{y}}_{c\; 3}^{*}(k)} \\{{\hat{y}}_{c\; 4}^{*}(k)}\end{bmatrix}},{K_{cdf} = \begin{bmatrix}K_{{cdf},1} \\K_{{cdf},2} \\K_{{cdf},3} \\K_{{cdf},4}\end{bmatrix}},{K_{cdd} = \begin{bmatrix}K_{{cdd},1} \\K_{{cdd},2} \\K_{{cdd},3} \\K_{{cdd},4}\end{bmatrix}}$

Compare the above desired constraint output response with the currentconstraint output response, yields,

$\begin{matrix}{{E_{\; c}{\overset{.}{v}(k)}} = {{K_{cRE}{{\hat{y}}_{c\;}^{*}(k)}} - \left( {{K_{cdf}F_{k}} + {K_{cdd}{\hat{d}(k)}}} \right) - \left( {{K_{cf}F_{k}} + {K_{cd}{\hat{d}(k)}}} \right)}} \\{= {{K_{cRE}{{\hat{y}}_{c\;}^{*}(k)}} - {\left( {K_{cdf} + k_{cf}} \right)F_{k}} - {\left( {K_{cdd} + K_{cd}} \right){\hat{d}(k)}}}} \\{= {{K_{cRE}{{\hat{y}}_{c\;}^{*}(k)}} + {K_{cF}F_{k}} + {K_{cD}{\hat{d}(k)}}}}\end{matrix}$ $\begin{matrix}{{\begin{bmatrix}e_{c\; 11} & e_{c\; 12} & e_{c\; 13} \\e_{c\; 21} & e_{c\; 22} & e_{c\; 23} \\e_{c\; 31} & e_{c\; 32} & e_{c\; 33} \\e_{c\; 41} & e_{c\; 42} & e_{c\; 43}\end{bmatrix}\begin{bmatrix}{{\overset{.}{v}}_{1}(k)} \\{{\overset{.}{v}}_{2}(k)} \\{{\overset{.}{v}}_{3}(k)}\end{bmatrix}} = {\begin{bmatrix}{K_{{ce}\; 1}{\hat{y}}_{c\; 1}^{*}} \\{K_{{ce}\; 2}{\hat{y}}_{c\; 2}^{*}} \\{K_{{ce}\; 3}{\hat{y}}_{c\; 3}^{*}} \\{K_{{ce}\; 4}{\hat{y}}_{c\; 4}^{*}}\end{bmatrix} + \begin{bmatrix}{{K_{cF}\left( {1,:} \right)}F_{k}} \\{{K_{cF}\left( {2,:} \right)}F_{k}} \\{{K_{cF}\left( {3,:} \right)}F_{k}} \\{{K_{cF}\left( {4,:} \right)}F_{k}}\end{bmatrix} +}} \\{\begin{bmatrix}{{K_{cD}\left( {1,:} \right)}{\hat{d}}_{k}} \\{{K_{cD}\left( {2,:} \right)}{\hat{d}}_{k}} \\{{K_{cD}\left( {3,:} \right)}{\hat{d}}_{k}} \\{{K_{cD}\left( {4,:} \right)}{\hat{d}}_{k}}\end{bmatrix}} \\{= \begin{bmatrix}{M_{p,k}(1)} \\{M_{p,k}(2)} \\{M_{p,k}(3)} \\{M_{p,k}(4)}\end{bmatrix}}\end{matrix}$

The decoupling matrix E_(c) between y_(c) and the pseudo input {dot over(v)} is derived based on the constraint controlled plant shaped by theprimary control, using generic form,

${E_{c}{\overset{.}{v}(k)}} = {{{M_{p}\begin{bmatrix}e_{c\; 11} & e_{c\; 12} & e_{c\; 13} \\e_{c\; 21} & e_{c\; 22} & e_{c\; 23} \\e_{c\; 31} & e_{c\; 32} & e_{c\; 33} \\e_{c\; 41} & e_{c\; 42} & e_{c\; 43}\end{bmatrix}}\begin{bmatrix}{\overset{.}{v}}_{1} \\{\overset{.}{v}}_{2} \\{\overset{.}{v}}_{3}\end{bmatrix}} = \begin{bmatrix}M_{p\; 1} \\M_{p\; 2} \\M_{p\; 3} \\M_{p\; 4}\end{bmatrix}}$

Physics Based Constraint Subset Classification

Based on plant (for example, engine) dynamics knowledge learned fromcycle partial studies and tests, assume that there are two subsets forthe concerned constraints based on effective control modes,corresponding to the two primary control outputs to be traded off,respectively:

y ₁-subset:{y _(c1) ,y _(c2)}

y ₂-subset:{y _(c3) ,y _(c4)}

Based on the above defined problem, the constraint control decouplingconstraints from the non-traded off controlled output y₃ is:

$\begin{matrix}{{\begin{bmatrix}e_{c\; 11} & e_{c\; 12} \\e_{c\; 21} & e_{c\; 22} \\e_{c\; 31} & e_{c\; 32} \\e_{c\; 41} & e_{c\; 42}\end{bmatrix}\begin{bmatrix}{\overset{.}{v}}_{1} \\{\overset{.}{v}}_{2}\end{bmatrix}} = {M_{p} - {{E_{c}\left( {:{,3}} \right)} \cdot {\overset{.}{v}}_{3}}}} \\{= {\begin{bmatrix}M_{p\; 1} \\M_{p\; 2} \\M_{p\; 3} \\M_{p\; 4}\end{bmatrix} - {\begin{bmatrix}e_{c\; 13} \\e_{c\; 23} \\e_{c\; 33} \\e_{c\; 43}\end{bmatrix}{\overset{.}{v}}_{3}}}} \\{= \begin{bmatrix}{M_{p}^{d_{3}}(1)} \\{M_{p}^{d_{3}}(2)} \\{M_{p}^{d_{3}}(3)} \\{M_{p}^{d_{3}}(4)}\end{bmatrix}} \\{= M_{p}^{d_{3}}}\end{matrix}$

Constraint Controller Set Determined by Constraint Subset

The physics-based classification of the above constraints determines notonly the constraint controller subsets but also the primary controltraded-off target for each subset. The details are: (a) The constraintsclassification should be physics-based, that is, for a given primarycontrol output, the projection of each constraint in its associatedconstraint subset along the primary control output dimension (ordirection) should be the dominant part of the constraint. In laymanwords, with respect to a given primary control reference, the (b) Thetotal number of constraint subsets is less than or equal to the primarycontrol handles; (c) The constraints in each subset are only to bemapped to one specified primary control trade-off target.

Decoupled SISO Constraint Controllers

Without loss of generality and clear formulation, assume that theconcerned constraints are active either in single subset only or in twosubsets at same time, and they are classified as:

y ₁-subset:{y _(c1) ,y _(c2) },y ₂-subset:{y _(c3) ,y _(c4)}.

All possible cases may be the following:

{y _(c1) },{y _(c2)},{y_(c3)},{y_(c4)},{y_(c1) ,y _(c3) },{y _(c1) ,y_(c4) },{y _(c2) ,y _(c3) },{y _(c2) ,y _(c4)}

It follows that at same time, the following constraint controllers needto be run in parallel.

Constraint controllers for two subset cases are derived below:

${\begin{bmatrix}e_{c\; 11} & e_{c\; 12} \\e_{c\; 31} & e_{c\; 32}\end{bmatrix}\begin{bmatrix}{\overset{.}{v}}_{1} \\{\overset{.}{v}}_{2}\end{bmatrix}} = {{{\begin{bmatrix}{M_{p}^{d_{3}}(1)} \\{M_{p}^{d_{3}}(3)}\end{bmatrix}\begin{bmatrix}e_{c\; 11} & e_{c\; 12} \\e_{c\; 41} & e_{c\; 42}\end{bmatrix}}\begin{bmatrix}{\overset{.}{v}}_{1} \\{\overset{.}{v}}_{2}\end{bmatrix}} = {{{\begin{bmatrix}{M_{p}^{d_{3}}(1)} \\{M_{p}^{d_{3}}(4)}\end{bmatrix}\begin{bmatrix}e_{c\; 21} & e_{c\; 22} \\e_{c\; 31} & e_{c\; 32}\end{bmatrix}}\begin{bmatrix}{\overset{.}{v}}_{1} \\{\overset{.}{v}}_{2}\end{bmatrix}} = {{{\begin{bmatrix}{M_{p}^{d_{3}}(2)} \\{M_{p}^{d_{3}}(3)}\end{bmatrix}\begin{bmatrix}e_{c\; 21} & e_{c\; 22} \\e_{c\; 41} & e_{c\; 42}\end{bmatrix}}\begin{bmatrix}{\overset{.}{v}}_{1} \\{\overset{.}{v}}_{2}\end{bmatrix}} = \begin{bmatrix}{M_{p}^{d_{3}}(2)} \\{M_{p}^{d_{3}}(4)}\end{bmatrix}}}}$

Define

${\left( E_{c}^{1,3} \right)^{- 1} = {\begin{bmatrix}e_{c\; 11} & e_{c\; 12} \\e_{c\; 31} & e_{c\; 32}\end{bmatrix}^{- 1} = {\begin{bmatrix}{i\; e_{c\; 11}^{1,3}} & {i\; e_{c\; 12}^{1,3}} \\{i\; e_{c\; 21}^{1,3}} & {i\; e_{c\; 22}^{1,3}}\end{bmatrix} = {i\; E_{c}^{1,3}}}}},{\left( E_{c}^{1,4} \right)^{- 1} = {\begin{bmatrix}e_{c\; 11} & e_{c\; 12} \\e_{c\; 41} & e_{c\; 42}\end{bmatrix}^{- 1} = {\begin{bmatrix}{i\; e_{c\; 11}^{1,4}} & {i\; e_{c\; 12}^{1,4}} \\{i\; e_{c\; 21}^{1,4}} & {i\; e_{c\; 22}^{1,4}}\end{bmatrix} = {i\; E_{c}^{1,4}}}}},{\left( E_{c}^{2,3} \right)^{- 1} = {\begin{bmatrix}e_{c\; 21} & e_{c\; 22} \\e_{c\; 31} & e_{c\; 32}\end{bmatrix}^{- 1} = {\begin{bmatrix}{i\; e_{c\; 11}^{2,3}} & {i\; e_{c\; 12}^{2,3}} \\{i\; e_{c\; 21}^{2,3}} & {i\; e_{c\; 22}^{2,3}}\end{bmatrix} = {i\; E_{c}^{2,3}}}}},{\left( E_{c}^{2,4} \right)^{- 1} = {\begin{bmatrix}e_{c\; 21} & e_{c\; 22} \\e_{c\; 41} & e_{c\; 42}\end{bmatrix}^{- 1} = {\begin{bmatrix}{i\; e_{c\; 11}^{2,4}} & {i\; e_{c\; 12}^{2,4}} \\{i\; e_{c\; 21}^{2,4}} & {i\; e_{c\; 22}^{2,4}}\end{bmatrix} = {i\; E_{c}^{2,4}}}}},$

Then the constraint controllers that further decouples constraints fromone another for two subset cases are:

{dot over (v)} ₁ ^(i,j) =ie _(c11) ^(i,j) ·M _(p) ^(d) ³ (i)+ie _(c12)^(i,j) ·M _(p) ^(d) ³ (j),

{dot over (v)} ₂ ^(i,j) =ie _(c21) ^(i,j) ·M _(p) ^(d) ³ (i)+ie _(c22)^(i,j) ·M _(p) ^(d) ³ (j).

Where i=1,2; j=3,4.

Constraint controllers for single subset cases are derived below:

$\begin{matrix}{{\begin{bmatrix}e_{c\; 11} \\e_{c\; 21}\end{bmatrix}{\overset{.}{v}}_{1}} = {{M_{p}^{d_{3}}\left( {1:2} \right)} - {{E_{c}\left( {{1:2},2} \right)} \cdot {\overset{.}{v}}_{2}}}} \\{= {\begin{bmatrix}{M_{p}^{d_{3}}(1)} \\{M_{p}^{d_{3}}(2)}\end{bmatrix} - \begin{bmatrix}{e_{c\; 12} \cdot {\overset{.}{v}}_{2}} \\{e_{c\; 22} \cdot {\overset{.}{v}}_{2}}\end{bmatrix}}} \\{{= M_{p,d_{2}}^{d_{3}}},}\end{matrix}$ $\begin{matrix}{{\begin{bmatrix}e_{c\; 32} \\e_{c\; 42}\end{bmatrix}{\overset{.}{v}}_{2}} = {{M_{p}^{d_{3}}\left( {3:4} \right)} - {{E_{c}\left( {{3:4},1} \right)} \cdot {\overset{.}{v}}_{1}}}} \\{= {\begin{bmatrix}{M_{p}^{d_{3}}(3)} \\{M_{p}^{d_{3}}(4)}\end{bmatrix} - \begin{bmatrix}{e_{c\; 31} \cdot {\overset{.}{v}}_{1}} \\{e_{c\; 41} \cdot {\overset{.}{v}}_{1}}\end{bmatrix}}} \\{{= M_{p,d_{1}}^{d_{3}}},}\end{matrix}$

Then the constraint controllers for single subset cases:

{dot over (v)} ₁ ^(i)=(e _(ci,1))⁻¹ ·M _(p,d) ₂ ^(d) ³ (i),

{dot over (v)} ₂ ^(j)=(e _(cj,2))⁻¹ ·M _(p,d) ₂ ^(d) ³ (i).

where i=1,2; j=3,4.

The MIMO constraint control design method demonstrated for twoconstraint subsets above is generic, and it can be easily applied to thecases where constraint subsets are more than two.

Constraint Control Selection Logic

For the problem of multiple variable control with multiple constraints,in general, the multiple constraints can be distributed in two or higherdimensions, i.e., there are two or more subsets of constraints. Further,the to-be-active constraints may be sometimes in one subset only andsometimes in two or more subsets at same time.

Therefore, it is desired that the Selection Logic should process allsubsets at each step and the transitions of multi-subsets and singlesubset. Specifically, in each subset, the Selection Logic selects themulti-subset most limiting constraint from the pseudo inputs resultedfrom two or more subsets active cases and the single subset mostlimiting constraint from the pseudo inputs resulted from single subsetactive case. Then the Selection Logic selects the most limitingconstraint from the multi-subset most limiting constraint, the singlesubset most limiting constraint, and the pseudo input generated by thetraded-off controlled output based on pre-determined selection logic,which is determined by the physical relationships between the max/minconstraints and the traded-off controlled output. Then in system level,i.e., considering the results from all subsets, the Selection Logicconducts integrated selection to make final decisions which pseudoinputs should be placed to the pseudo input entries.

Considering for a given constraint, single subset case and multi-subsetscase cannot happen to it at same time, however it can transition fromone to the other, therefore, in a given subset, the single subset caseand multi-subsets case need to go through separate selection processes,and the transition will naturally go through by the selection resultsfrom system level integrated selection.

An example for demonstration of the selection logic is provided below.The example is associated with y₁ and y₂ respectively, and each subsethas 2 constraints: y₁-subset: {max y_(c1), min y_(c1), min y_(c2)},y₂-subset: {max y_(c3), min y_(c3), max y_(c4)}.

Assume that there are two subsets of the concerned constraints,corresponding to two primary control outputs to be traded off,respectively:

y ₁-subset:{y _(c1) ,y _(c2)}

y ₂-subset:{y _(c3) ,y _(c4)}i.e.,{max y _(c3),min y _(c3),max y _(c4)}

and consider {max y_(c1), min y_(c1), min y_(c2)} and {max y_(c3), miny_(c3), max y_(c4)} case.

Without loss of generality and clear formulation, assume that theconcerned constraints are active either in single subset only or in twosubsets at same time, all possible cases are given below:

{y _(c1) },{y _(c2)},{y_(c3)},{y_(c4)},{y_(c1) ,y _(c3) },{y _(c1) ,y_(c4) },{y _(c2) ,y _(c3) },{y _(c2) ,y _(c4)},

Considering the constraints {max y_(c1), min y_(c1), min y_(c2)} iny₁-subset, assume that to satisfy max y_(c1) needs to reduce y₁, i.e.,{dot over (v)}₁ ^(y) ¹ <0, if max y_(c1) is violated, it generates {dotover (v)}₁ ^(1+,•)<0 or, {dot over (v)}₁ ¹⁺<0, therefore, select theminimum value from {dot over (v)}₁ ^(1+,•), {dot over (v)}₁ ¹⁺, and {dotover (v)}₁ ^(y) ¹ can satisfy max y_(c1); to satisfy min y_(c1) needs toincrease y₁, i.e., {dot over (v)}₁ ^(y) ¹ >0, if min y_(c1) is violated,it generates {dot over (v)}₁ ^(1−,•)>0 or, {dot over (v)}₁ ¹⁻>0; tosatisfy min y_(c2) needs to increase y₁, i.e., {dot over (v)}₁ ^(y)¹ >0, if min y_(c2) is violated, it generates {dot over (v)}₁ ^(2−,•)>0or, {dot over (v)}₁ ²⁻>0; therefore, select the maximum value from {dotover (v)}₁ ^(1−,•), {dot over (v)}₁ ^(2−,•), {dot over (v)}₁ ¹⁻, {dotover (v)}₁ ²⁻ and {dot over (v)}₁ ^(y) ¹ can satisfy both min y_(c1) andmin y_(c2). Also assume that maximum constraint overrules the minimumconstraints.

FIG. 4 demonstrates the assumed y₁-subset selection logic for {maxy_(c1), min y_(c1), min y_(c2)}. In single subset case, min y_(c1) andmin y_(c2) generate {dot over (v)}₁ ¹⁻ and {dot over (v)}₁ ²⁻,respectively; max y_(c1) generates {dot over (v)}₁ ¹⁺. Applying theassumed relationships for y₁-subset aforementioned, the single subsetmost limiting constraint is {dot over (v)}_(1,S) ^(ML)=min(max({dot over(v)}₁ ¹⁻,{dot over (v)}₁ ²⁻),{dot over (v)}₁^(1+). In multi-subsets case, min y) _(c1) generates {dot over (v)}₁¹⁻³⁺, {dot over (v)}₁ ¹⁻³⁻ and {dot over (v)}₁ ¹⁻ ⁴⁺, min y_(c2)generates {dot over (v)}₁ ²⁻³⁺, {dot over (v)}₁ ²⁻³⁻ and {dot over (v)}₁²⁻⁴⁺, max y_(c1) generates {dot over (v)}₁ ¹⁺³⁺, {dot over (v)}₁ ¹⁺³⁻i,and {dot over (v)}₁ ¹⁺⁴⁺. Applying the assumed relationshipsaforementioned, the multi-subsets most limiting constraint of is {dotover (v)}_(1,M) ^(ML)=min(max({dot over (v)}₁ ¹⁻³⁺, {dot over (v)}₁¹⁻³⁻,{dot over (v)}₁ ¹⁻⁴⁺, {dot over (v)}₁ ²⁻³⁺, {dot over (v)}₁ ²⁻³⁻,{dot over (v)}₁ ²⁻⁴⁺), {dot over (v)}₁ ¹⁺³⁺, {dot over (v)}₁ ¹⁺³⁻, {dotover (v)}₁ ¹⁺⁴⁺); the y₁-subset most limiting constraint is: {dot over(v)}₁ ^(ML)=min({dot over (v)}₁ ^(y) ¹ , {dot over (v)}_(1,S) ^(ML),{dot over (v)}_(1,M) ^(ML)).

Considering the constraints {max y_(c3), min y_(c3), max y_(c24)} iny₂-subset assume that to satisfy max y_(c3) needs to increase y₂, i.e.,{dot over (v)}₂ ^(y) ² ≦0, if max y_(c3) is violated, it generates{circumflex over (v)}₂ ^(•,3+)>0 or, {dot over (v)}₂ ³⁺>0; to satisfymax y_(c4) needs to increase y₂, i.e., {dot over (v)}₂ ^(y) ^(2>) 0, ifmax y_(c4) is violated, it generates {dot over (v)}₂ ^(•,4+)>0 or, {dotover (v)}₂ ⁴⁺>0; therefore, select the maximum value from {dot over(v)}₂ ^(•3+), {dot over (v)}₂ ^(•,4+), {dot over (v)}₂ ³⁺, {dot over(v)}₂ ⁴⁺ and {dot over (v)}₂ ^(y) ² can satisfy both max y_(c3) and maxy_(c4); to satisfy min y_(c3) needs to reduce y₂, i.e., {dot over (v)}₂^(y) ² <0, if min y_(c3) is violated, it generates {dot over (v)}₂^(•,3−)<0 or, {dot over (v)}₂ ³⁻<0, therefore, select the minimum valuefrom {dot over (v)}₂ ^(•,3−), {dot over (v)}₂ ³⁻, and {dot over (v)}₂^(y) ² can satisfy min y_(c3). Also assume that maximums overrule theminimum constraint.

FIG. 5 demonstrates the assumed y₂-subset selection logic for {maxy_(c3), min y_(c3), max y_(c24)}. In single subset case, max y_(c3) andmax y_(c4) generate {dot over (v)}₂ ³⁺ and {dot over (v)}₂ ⁴⁺,respectively; min y_(c3) generates {dot over (v)}₂ ³⁻. Applying theassumed relationships for y₂-subset aforementioned, the single subsetmost limiting constraint is {dot over (v)}_(2,S) ^(ML)=max(max({dot over(v)}₂ ³⁺, {dot over (v)}₂ ⁴⁺), {dot over (v)}₂ ³⁻). In multi-subsetscase, max y_(c3) generates {dot over (v)}₂ ¹⁻³⁺, {dot over (v)}₂ ¹⁺³⁺and {dot over (v)}₂ ²⁻³⁺, max y_(c4) generates {dot over (v)}₂ ¹⁻⁴⁺,{dot over (v)}₂ ¹⁺⁴⁺ and {dot over (v)}₂ ²⁻⁴⁺, min y_(c3) generates {dotover (v)}₂ ¹⁺³⁻, {dot over (v)}₂ ¹⁻³⁻, and {dot over (v)}₂ ²⁻³⁻.Applying the assumed relationships aforementioned, the multi-subsetsmost limiting constraint is {dot over (v)}_(2,S) ^(ML)=max(min({dot over(v)}₂ ¹⁺³⁻, {dot over (v)}₂ ¹⁻³⁻, {circumflex over (v)}₂ ²⁻³⁻), {dotover (v)}₂ ¹⁺³⁺, {dot over (v)}₂ ¹⁻³⁺, {dot over (v)}₂ ¹⁻⁴⁺, {dot over(v)}₂ ¹⁺⁴⁺, {dot over (v)}₂ ²⁻³⁺, {dot over (v)}₂ ²⁻⁴⁺); the y₂-subsetmost limiting constraint is: {dot over (v)}₂ ^(ML)=max({dot over (v)}₂^(y) ² , {dot over (v)}_(2,S) ^(ML), {dot over (v)}_(2,M) ^(ML)).

In system level, considering both subsets y₁-subset and y₂-subset areactive at one time, one of them is active at another time, and thetransitions, the integrated selection makes decisions based on theresults from the two subsets: {dot over (v)}₁ ^(ML), {dot over (v)}₁^(y) ¹ , {dot over (v)}_(1,S) ^(ML), {dot over (v)}_(1,M) ^(ML), {dotover (v)}₂ ^(ML), {dot over (v)}₂ ^(y) ² , {dot over (v)}_(2,S) ^(ML),{dot over (v)}_(2,M) ^(ML), and desires to make smooth transitions.

FIG. 6 demonstrates the integrated selection logic. The checkingconditions are determined by the results from the two subsets: {dot over(v)}₁ ^(ML), {dot over (v)}₁ ^(y) ¹ , {dot over (v)}_(1,S) ^(ML), {dotover (v)}_(1,M) ^(ML), {dot over (v)}₂ ^(ML), {dot over (v)}₂ ^(y) ² ,{dot over (v)}_(2,S) ^(ML), {dot over (v)}_(2,M) ^(ML).

Step 1: If both subsets are active, i.e., the first condition is true,then both y₁ and y₂ need to be traded off, it follows that {dot over(v)}₁ ^(y) ¹ is replaced by {dot over (v)}_(1,M) ^(ML), and {dot over(v)}₂ ^(y) ² replaced by {dot over (v)}_(2,M) ^(ML).

Step 2: If not both subsets are active, check condition 2—if y₁ subsetis active only. If condition 2 is true, then this is single subset case,y₁ needs to be traded off, and y₂ need to stay, it follows that {dotover (v)}₁ ^(y) ¹ is replaced by {dot over (v)}_(1,S) ^(ML), and {dotover (v)}₂ ^(y) ² is kept.

Step 3: If condition 2 is not true, check condition 3—if y₂ subset isactive only. If condition 3 is true, then this is single subset case, y₂needs to be traded off, and y₁ need to stay, it follows that {dot over(v)}₂ ^(y) ² is replaced by {dot over (v)}_(2,S) ^(ML), and {dot over(v)}₁ ^(y) ¹ is kept.

Step 4: If condition 3 is not true, then there are no active constraintsin both subsets, it follows that {dot over (v)}₁ ^(y) ¹ and {dot over(v)}₂ ^(y) ² stay, no primary controlled outputs are traded off.

More specifically, referring to FIG. 6, the algorithm starts at step 20and proceeds to step 22. At step 22, if {dot over (v)}₁ ^(y) ¹ ≠{dotover (v)}₁ ^(ML) && {dot over (v)}₂ ^(y) ² ≠{dot over (v)}₂ ^(ML)

Then, at step 24,

{dot over (v)} ₁ ={dot over (v)} ₁ ^(ML);

{dot over (v)} ₂ ={dot over (v)} ₂ ^(ML);

then proceed to end at step 26.

Otherwise the algorithm proceeds to step 28. At step 28, if {dot over(v)}₁ ^(y) ¹ ={dot over (v)}₁ ^(ML) && {dot over (v)}₂ ^(y) ² ≠{dot over(v)}₂ ^(ML)

Then, at step 30,

{dot over (v)}₁={dot over (v)}₁ ^(y) ¹

{dot over (v)}₂={dot over (v)}_(2,S) ^(ML);

then proceed to end at step 26.

Otherwise the algorithm proceeds to step 32. At step 32, if {dot over(v)}₁ ^(y) ¹ ≠{dot over (v)}₁ ^(ML) && {dot over (v)}₂ ^(y) ² ={dot over(v)}₂ ^(ML)

Then, at step 34,

{dot over (v)} ₁ ={dot over (v)} _(1,S) ^(ML);

{dot over (v)} ₂ ={dot over (v)} ₂ ^(y) ² ;

then proceed to end at step 26.

Otherwise, the algorithm proceeds to step 36, where,

{dot over (v)}₁={dot over (v)}₁ ^(y) ¹ ;

{dot over (v)}₂={dot over (v)}₂ ^(y) ² ;

End at step 26.

The architecture of a generic Advanced Multiple Variable Control withHigh Dimension Multiple Constraints is shown in FIG. 1. It works in theway described below:

(1) A multiple input multiple output (MIMO) primary decouplingcontroller 40: (a) generates control command derivatives—{dot over (u)}to the integral action 42; and (b) provides decoupled control (usingdynamics inversion or some other known method) from {dot over (v)} to y.The dynamics of the decoupled controlled plant (from {dot over (v)} toy) are shaped to enable desired robust control of the primary controloutputs. The coupled I/O mapping between {dot over (u)} and y becomesdecoupled new I/O mapping between {dot over (v)} and y, and the pseudoinput entries {dot over (v)} provide the common comparable points thatthe pseudo inputs generated by constraint controllers can be comparedwith the pseudo inputs generated by primary controlled outputs inaccordance with the Selection Logic 50.

(2) A set of SISO lead/lag controllers 52 may be installed upstream ofthe primary MIMO Primary Decoupling Controller 40 to extend thebandwidths of decoupled primary SISO control loops, providing {dot over(v)}* to the primary MIMO Primary Decoupling Controller 40. Because thisis a common element that would affect the primary and the constraintcontrol, this would also extend the SISO closed loop bandwidth whenrunning to constraints.

(3) A set of decoupled SISO controllers 56 for controlled outputstracking which receive primary controlled output tracking errors(Control References (58) minus the Controlled Outputs (48)) and providedesired primary controlled output based pseudo inputs {dot over(v)}_(p), respectively. Tunes the primary control loops independently ofthe constraint outputs, so they can be optimized without impacting thecharacteristics of the constraint control.

(4) A multiple input multiple output (MIMO) constraint decouplingcontroller 60 that controls the New Controlled Plant formed by the MIMOprimary controller 40 and the physical plant 62. The MIMO constraintdecoupling controller 60: (a) generates pseudo inputs {dot over (v)}_(c)based on the desired constraint responses for the Selection Logic; and(b) decouples the constraints from one another based on the newly shapedcontrolled plant {dot over (v)} to y_(c); and (c) decouples theconstraints from the non-traded off primary controlled outputs byrejecting the non-traded off primary controlled outputs as knowndisturbance inputs {dot over (v)}_(pr).

(5) A set of decoupled SISO controllers 64 for constraint outputstracking which receive constraint output tracking errors (ConstraintLimits minus Constraint Outputs) and provide desired constraincontrolled pseudo inputs {dot over (v)}_(c), respectively. Tunes theconstraint control loops independently of the primary outputs, so theycan be optimized without impacting the characteristics of the primarycontrol.

(6) Selection Logic 50 compares the pseudo inputs generated by everygiven subset of constraints and the pseudo input generated by theprimary controlled output associated with that subset, selects the mostlimiting constraint for each subset, and makes system level selectionintegration to determine the final pseudo inputs to go into the SISOLead/Lag and MIMO Primary Decoupling Controller.

(7) An integral action 42 includes a set of integrators common to boththe primary and constraint control. The integral action integrates each{dot over (u)} into a corresponding u thus forming the reference foreach actuator inner loop. Each integrator may be dynamically limited torespect corresponding actuator operating limits. The integrator is shownin FIG. 3.

As an example implementation, architecture of a 3×3 Advanced MultipleVariable Control with Two sets of Constraints is shown in FIG. 2 (theelements in FIG. 2 directly or indirectly corresponding to elements inFIG. 1 have the same numerals, but have added one-hundred). It works inthe way described below:

(1) 3×3 MIMO primary decoupling controller 140 not only generatingcontrol command derivatives—{dot over (u)}₁, {dot over (u)}₂ and {dotover (u)}₃ to the integral action but also shaping the common pseudoinput entries—{dot over (v)}₁, {dot over (v)}₂, and {dot over (v)}₃based on primary controlled output y (y₁, y₂, y₃) to {dot over (u)}({dot over (u)}₁, {dot over (u)}₂, {dot over (u)}₃) dynamics inversionand decoupled primary SISO desired plant dynamics via state feedbacksuch that the coupled I/O mapping between {dot over (u)} and Y becomesdecoupled new I/O mapping between {dot over (v)} and Y with desiredplant dynamics, and the pseudo input entries {dot over (v)} provide thecommon comparable points that the pseudo inputs generated by constraintcontrol can be compared with the pseudo inputs generated by primarycontrol accordingly in Selection Logic.

(2) There are only three control handles in this example, which impliesthere will be at most three subsets of constraints. In this case, assumetwo subsets of constraints associated with y₁ and y₂ respectively, andeach subset has two constraints: y₁-subset: {max y_(c1), min y_(c1), miny_(c2)}, y₂-subset: {max y_(c3), min y_(c3), max y_(c4)}.

(3) Three SISO lead/lag controllers 152 that is intend to extend thebandwidths of decoupled primary SISO control loops, respectively, alsoare common to be used by the selected constraints for the same purpose.If the SISO lead/lags are not needed, then they can be set to 1,respectively.

(4) Three decoupled SISO proportional controllers 156 for controlledoutputs tracking which receive primary controlled output tracking errors(Control References minus Controlled Outputs) and provide desiredprimary controlled outputs based pseudo inputs—{dot over (v)}₁, {dotover (v)}₂, and {dot over (v)}₃, respectively. Assumed y₃ is not to betraded off, {dot over (v)}₃ will serve as known disturbance input inconstraint control; {dot over (v)}₁ and {dot over (v)}₂ will go intoSelection Logic 150 to be compared with the pseudo inputs generated byconstraint controllers.

(5) A set of 2×2 MIMO constraint decoupling controllers 160 is shown(2×2 is assumed case in this example, any number of such constraintdecoupling controllers can be utilized, depending upon the actualcontrol system) to control two subsets of constraints. The constraintdecoupling controllers in this example decouple the constraint of subset1 from constraint of subset 2, respectively, i.e., decoupling y_(c1)from y_(c3), y_(c1) from y_(c4), y_(c2) from y_(c3), y_(c2) from y_(c4),based on constraints y_(c) to {dot over (v)} dynamics inversion anddecoupled constraint SISO desired dynamics via state feedback, anddecoupling the constraints from the non-traded off primary controlledoutput y₃ by rejecting the known disturbance inputs {dot over (v)}₃, itfollows that the resulted four decoupled SISO constraint controllers((y_(ci)->{dot over (v)}₁ ^(•,j)), i=1,2; j=3, 4) for generating pseudoinputs {dot over (v)}₁ ^(i,j) for subset 1 to be compared with {dot over(v)}₁ and four decoupled SISO constraint controllers ((y_(cj)->{dot over(v)}₂ ^(i,•)), i=1, 2; j=3, 4) for generating pseudo inputs {dot over(v)}₂ ^(i,j) for subset 2 to be compared with {dot over (v)}₂. If aconstraint has two limits, then the same decoupled SISO constraintcontroller generates two outputs, that is, two pseudo inputs eachcorresponds to one limit input; if the constraint has one limit, thenthe decoupled SISO constraint controller generates one output, that is,one pseudo input. Based on assumptions on constraints in this case,there are following generated pseudo inputs, {dot over (v)}₁ ^(1+,3−),{dot over (v)}₁ ^(1+,3+), {dot over (v)}₁ ^(1−,3+), {dot over (v)}₁^(1−,3−), {dot over (v)}₁ ^(1+,4+), {dot over (v)}₁ ^(1−,4+), {dot over(v)}₁ ^(2−,3−), {dot over (v)}₁ ²⁺⁻³⁺, {dot over (v)}₁ ²⁻⁴⁺, in subset 1to be compared with {dot over (v)}₁; and {dot over (v)}₂ ^(1+,3−), {dotover (v)}₂ ^(1+,3+), {dot over (v)}₂ ^(1−,3+), {dot over (v)}₂ ^(1−,3−),{dot over (v)}₂ ^(1+,4+), {dot over (v)}₂ ^(1−,4+), {dot over (v)}₂^(2−,3−), {dot over (v)}₂ ²⁺⁻³⁺, {dot over (v)}₁ ^(2−,4+) in subset 2 tobe compared with

(6) Four decoupled SISO controllers 164 for constraint outputs trackingwhich receive constraint output tracking errors (lim y_(ci)=y_(ci), i=1,2, 3, 4) and shape desired constraint tracking responses, respectively;same constraint with different limits as reference inputs will use thesame constraint decoupled SISO controller.

(7) One selection logic 150 for subset 1 and one selection logic 150 forsubset 2. Each compares the pseudo inputs generated by the given subsetof constraints and the pseudo input generated by the primary controlledoutput associated with that subset, selects the most limiting constraintfor each subset, and determines the final pseudo input to go into theSISO Lead/Lag 152 and MIMO Primary Decoupling Controller 140.

(8) Three common SISO integrators 142 which are not individually shownin FIG. 2 due to limited space. Each integrator works for each decoupledprimary SISO loop, generates each {dot over (u)}_(i) from each {dot over(u)}_(i), i=1, 2, 3, and passes it to each control handle—actuator innerloop as input command reference accordingly, each integrator isdynamically saturated with the max/min operating range of a givenactuator. The integrator 142 is shown in FIG. 3.

The common SISO integrator 142 is shown in FIG. 3. (1) Calculate: {dotover (u)}k={tilde over ({dot over (u)}_(k)+{dot over (u)}k−1 accordingto the perturbation definition; (2) Apply the max/min operating ratelimits to {dot over (u)}_(k); (3) Calculate current step command change:Δu_(k)=T_(s)·{dot over (u)}_(k), where T_(s) is the sampling time; (4)Calculate current step command: u_(k)=Δu_(k)+u_(k−1); (5) Apply themax/min operating limits {dot over (u)}_(k) as shown in FIG. 4.

Technically, the current approach overcomes the fundamental,longstanding MIMO mode selection challenge of selecting between multiplesets of control modes due to the coupled and confounded set of inputvariables associated with a coupled complex plant process (for example,a typical gas turbine engine processes). The pseudo inputs from the newcontrolled plant resulted from the primary control provide MIMO modeselection criteria which have a direct, one-to-one correspondence from agiven constraint to the specific performance trade-off decisionaccording to pre-determined rules. This solution preserves SISO-typemode selection simplicity even with high dimension constraints systems.It allows a simple SISO constraint controller or certain simple SISOconstraint controllers, selected from multiple constraints, thatreconfigure the existing primary MIMO control online by replacing thetraded-off output with the selected constraint when a single subset isactive or replacing the traded-off outputs with the selected constraintswhen multiple subsets are active. The resultant design has explicitphysical meaning, is simple, deterministic, fundamentally robust, andeasily maintainable.

It is to be understood the control system architectures disclosed hereinmay be provided in any manner known to those of ordinary skill,including software solutions, hardware or firmware solutions, andcombinations of such. Such solutions would incorporate the use ofappropriate processors, memory (and software embodying any algorithmsdescribed herein may be resident in any type of non-transitory memory),circuitry and other components as is known to those of ordinary skill.

Having disclosed the inventions described herein by reference toexemplary embodiments, it will be apparent to those of ordinary skillthat alternative arrangements and embodiments may be implemented withoutdeparting from the scope of the inventions as described herein. Further,it will be understood that it is not necessary to meet any of theobjects or advantages of the invention(s) stated herein to fall withinthe scope of the inventions, because undisclosed or unforeseenadvantages may exist.

What is claimed is:
 1. A control system for a physical plant,comprising: an integral action control unit providing control signalsfor a physical plant; a multiple-input-multiple-output (MIMO) primarydecoupling controller decoupling controlled outputs from one another andshaping pseudo inputs/controlled outputs desired plant dynamics andproviding control command derivatives to the integral action controlunit and thereby forming at least part of a new controlled plant; and amultiple-input-multiple-output (MIMO) constraint decoupling controllerdecoupling constraint outputs from one another and from the non-tradedoff controlled output(s) and shaping pseudo inputs/constraint outputsdesired plant dynamics and providing pseudo inputs to the new controlledplant.
 2. The control system of claim 1, further comprising a selectionlogic section for selecting pseudo inputs for the primary decouplingcontroller from the pseudo inputs calculated by the MIMO constraintdecoupling controller and the pseudo inputs calculated by the controlledoutput tracking controllers based on primary decoupling control.
 3. Thecontrol system of claim 2, further comprising a set of decoupledsingle-input-single-output (SISO) controlled output tracking controllersreceiving controlled output tracking error signals and providing pseudoinput signals to the new controlled plant.
 4. The control system ofclaim 3, wherein the selection logic compares the pseudo inputs from theMIMO constraint decoupling controller and the pseudo inputs from theSISO controlled output tracking controllers and selects the mostlimiting constraint for each primary SISO control loop and provides themto the (SISO) lead/lag controllers.
 5. The control system of claim 1,further comprising a set of single-input-single-output (SISO) lead/lagcontrollers to extend the bandwidths of decoupled primary SISO controlloops, providing v-dot-star to the primary decoupling controller.
 6. Thecontrol system of claim 1, wherein the MIMO constraint decouplingcontroller decouples the constraints from the non-traded off primarycontrolled outputs by rejecting non-traded off primary controlledoutputs as known disturbance inputs.
 7. The control system of claim 1,wherein the MIMO constraint decoupling controller decouples constraintoutputs from one another, and decouples the constraints from thenon-traded off primary controlled outputs.
 8. The control system ofclaim 1, further comprising a set of single-input-single-output (SISO)constraint output tracking controllers that receive constraint outputtracking errors from the physical plant and shape desired constraintresponses and provide the inputs to the MIMO constraint decouplingcontroller.
 9. The control system of claim 8, wherein the constraintoutput tracking errors are determined, at least in part, based upon thedifferences between predetermined constraint limits and constraintoutputs.
 10. A method for multiple variable control of a physical plantwith high dimensions multiple constraints, comprising the steps of:controlling a physical plant with multiple inputs and multiple primarycontrolled outputs and high dimension multiple constraints; decouplingthe multiple primary controlled outputs from one another and shapingpseudo inputs/controlled outputs desired plant dynamics; decoupling themultiple constraints from one another; decoupling the multipleconstraints from non-traded off primary controlled outputs; shapingpseudo inputs/constraint outputs desired plant dynamics; selecting themost limiting constraints for the pseudo input entries;
 11. The methodof claim 10, wherein the step of decoupling the multiple controlledoutputs involves a multi-input-multi-output (MIMO) primary decouplingcontroller.
 12. The method of claim 10, wherein the step of decouplingthe multiple constraints involves a multi-input-multi-output (MIMO)constraint decoupling controller.
 13. The method of claim 10, whereinthe step of selecting the most limiting constraints involves a selectionlogic.
 14. The method of claim 13, wherein the selection logic comparingthe pseudo inputs generated by given subsets of constraints and thepseudo inputs generated by the primary controlled outputs associatedwith those subset, respectively, and selecting the most limitingconstraint for each subset based, at least in part, on thosecomparisons.
 15. The method of claim 10, wherein the MIMO primarydecoupling controller provides decoupled control using dynamicsinversion.
 16. The method of claim 10, further comprising a step ofextending the bandwidths of decoupled primary control loops using a setof single-input-single-output (SISO) lead/lag controllers upstream ofthe MIMO primary decoupling controller.
 17. The method of claim 10,wherein the step of decoupling the multiple constraints from non-tradedoff primary controlled outputs includes a step of rejecting thenon-traded off primary controlled outputs as known disturbance inputs.18. A method for multiple variable control of a physical plant with highdimension multiple constraints, comprising the steps of: mathematicallydecoupling primary controlled outputs of a controlled physical plantfrom one another; mathematically decoupling constraints from oneanother; mathematically decoupling constraints from non-traded offprimary controlled outputs; and controlling the physical plant using thedecoupled non-traded off primary controlled outputs and the decoupledselected most limiting constraints.
 19. The method of claim 18, furthercomprising a step of selecting one or more most limiting constraints.20. The method of claim 10, wherein the Selection Logic determining themost limiting constraint for each subset based on pre-determined rulesand managing the constraint active/inactive transitions cross subsetsand providing smooth pseudo inputs to the MIMO primary decouplingcontroller.